TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, general topology is the branch of
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

that deals with the basic
set-theoretic illustrating the intersection of two sets. Set theory is a branch of mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topi ...
definitions and constructions used in topology. It is the foundation of most other branches of topology, including
differential topology In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
,
geometric topology In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, and
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
. Another name for general topology is point-set topology. The fundamental concepts in point-set topology are ''continuity'', ''compactness'', and ''connectedness'': *
Continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
s, intuitively, take nearby points to nearby points. *
Compact set In mathematics, more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed set, closed (i.e., containing all its limit points) and bounded set, bounded (i.e., having all ...
s are those that can be covered by finitely many sets of arbitrarily small size. *
Connected set In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric o ...
s are sets that cannot be divided into two pieces that are far apart. The terms 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using the concept of
open set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a ''topology''. A set with a topology is called a ''
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
''. ''
Metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
s'' are an important class of topological spaces where a real, non-negative distance, also called a ''
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...
'', can be defined on pairs of points in the set. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.

# History

General topology grew out of a number of areas, most importantly the following: *the detailed study of subsets of the
real line In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
(once known as the ''topology of point sets''; this usage is now obsolete) *the introduction of the
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

concept *the study of
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
s, especially
normed linear space The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk Dunkirk (, ; french: Dunkerque ; vls, label=French Flemish, Duunkerke; nl, Duinkerke(n) ) is a Communes of France, ...
s, in the early days of
functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
. General topology assumed its present form around 1940. It captures, one might say, almost everything in the intuition of continuity, in a technically adequate form that can be applied in any area of mathematics.

# A topology on a set

Let ''X'' be a set and let ''τ'' be a
family In human society A society is a Social group, group of individuals involved in persistent Social relation, social interaction, or a large social group sharing the same spatial or social territory, typically subject to the same Politic ...
of
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s of ''X''. Then ''τ'' is called a ''topology on X'' if: # Both the
empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, cha ...

and ''X'' are elements of ''τ'' # Any union of elements of ''τ'' is an element of ''τ'' # Any
intersection The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...
of finitely many elements of ''τ'' is an element of ''τ'' If ''τ'' is a topology on ''X'', then the pair (''X'', ''τ'') is called a ''topological space''. The notation ''Xτ'' may be used to denote a set ''X'' endowed with the particular topology ''τ''. The members of ''τ'' are called ''
open set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s'' in ''X''. A subset of ''X'' is said to be closed if its
complement A complement is often something that completes something else, or at least adds to it in some useful way. Thus it may be: * Complement (linguistics), a word or phrase having a particular syntactic role ** Subject complement, a word or phrase addi ...
is in ''τ'' (i.e., its complement is open). A subset of ''X'' may be open, closed, both (
clopen set upright=1.3, A Graph (discrete mathematics), graph with several clopen sets. Each of the three large pieces (i.e. connected component (topology), components) is a clopen set, as is the union of any two or all three. In topology, a clopen set (a po ...
), or neither. The empty set and ''X'' itself are always both closed and open.

## Basis for a topology

A base (or basis) ''B'' for a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
''X'' with
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
''T'' is a collection of
open set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s in ''T'' such that every open set in ''T'' can be written as a union of elements of ''B''. We say that the base ''generates'' the topology ''T''. Bases are useful because many properties of topologies can be reduced to statements about a base that generates that topology—and because many topologies are most easily defined in terms of a base that generates them.

## Subspace and quotient

Every subset of a topological space can be given the
subspace topologyIn topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a Topological_space#Definitions, topology induced from that of ''X'' called the subspace topology (or the relative ...

in which the open sets are the intersections of the open sets of the larger space with the subset. For any
indexed family In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
of topological spaces, the product can be given the
product topology Product may refer to: Business * Product (business) In marketing, a product is an object or system made available for consumer use; it is anything that can be offered to a market Market may refer to: *Market (economics) *Market economy *Mark ...
, which is generated by the inverse images of open sets of the factors under the projection mappings. For example, in finite products, a basis for the product topology consists of all products of open sets. For infinite products, there is the additional requirement that in a basic open set, all but finitely many of its projections are the entire space. A quotient space is defined as follows: if ''X'' is a topological space and ''Y'' is a set, and if ''f'' : ''X''→ ''Y'' is a
surjective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
, then the
quotient topology as the quotient space of a disk Disk or disc may refer to: * Disk (mathematics) * Disk storage Music * Disc (band), an American experimental music band * Disk (album), ''Disk'' (album), a 1995 EP by Moby Other uses * Disc (galaxy), a disc-sha ...
on ''Y'' is the collection of subsets of ''Y'' that have open
inverse image In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

s under ''f''. In other words, the quotient topology is the finest topology on ''Y'' for which ''f'' is continuous. A common example of a quotient topology is when an
equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
is defined on the topological space ''X''. The map ''f'' is then the natural projection onto the set of
equivalence class In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
es.

## Examples of topological spaces

A given set may have many different topologies. If a set is given a different topology, it is viewed as a different topological space.

### Discrete and trivial topologies

Any set can be given the
discrete topology In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...
, in which every subset is open. The only convergent sequences or nets in this topology are those that are eventually constant. Also, any set can be given the
trivial topologyIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
(also called the indiscrete topology), in which only the empty set and the whole space are open. Every sequence and net in this topology converges to every point of the space. This example shows that in general topological spaces, limits of sequences need not be unique. However, often topological spaces must be
Hausdorff space In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...

s where limit points are unique.

### Cofinite and cocountable topologies

Any set can be given the cofinite topology in which the open sets are the empty set and the sets whose complement is finite. This is the smallest T1 topology on any infinite set. Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or its complement is countable. When the set is uncountable, this topology serves as a counterexample in many situations.

### Topologies on the real and complex numbers

There are many ways to define a topology on R, the set of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s. The standard topology on R is generated by the . The set of all open intervals forms a
base Base or BASE may refer to: Brands and enterprises * Base (mobile telephony provider), a Belgian mobile telecommunications operator *Base CRM Base CRM (originally Future Simple or PipeJump) is an enterprise software company based in Mountain Vie ...
or basis for the topology, meaning that every open set is a union of some collection of sets from the base. In particular, this means that a set is open if there exists an open interval of non zero radius about every point in the set. More generally, the
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
s R''n'' can be given a topology. In the usual topology on R''n'' the basic open sets are the open
ball A ball is a round object (usually spherical of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a geometrical object in three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional s ...
s. Similarly, C, the set of
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s, and C''n'' have a standard topology in which the basic open sets are open balls. The real line can also be given the
lower limit topology In mathematics, the lower limit topology or right half-open interval topology is a topological space, topology defined on the set \mathbb of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and h ...
. Here, the basic open sets are the half open intervals [''a'', ''b''). This topology on R is strictly finer than the Euclidean topology defined above; a sequence converges to a point in this topology if and only if it converges from above in the Euclidean topology. This example shows that a set may have many distinct topologies defined on it.

### The metric topology

Every
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
can be given a metric topology, in which the basic open sets are open balls defined by the metric. This is the standard topology on any normed vector space. On a finite-dimensional vector space this topology is the same for all norms.

### Further examples

* There exist numerous topologies on any given finite set. Such spaces are called
finite topological space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s. Finite spaces are sometimes used to provide examples or counterexamples to conjectures about topological spaces in general. * Every
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

has a
natural topology In any domain of mathematics, a space has a natural topology if there is a topology on the space which is "best adapted" to its study within the domain in question. In many cases this imprecise definition means little more than the assertion that t ...
, since it is locally Euclidean. Similarly, every
simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given space. For e ...

and every
simplicial complex In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
inherits a natural topology from Rn. * The
Zariski topology In algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commut ...
is defined algebraically on the
spectrum of a ring In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
or an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematic ...
. On R''n'' or C''n'', the closed sets of the Zariski topology are the
solution set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s of systems of
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

equations. * A linear graph has a natural topology that generalises many of the geometric aspects of
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...
s with vertices and . * Many sets of
linear operator In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s in
functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function. * Any
local field In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
has a topology native to it, and this can be extended to vector spaces over that field. * The Sierpiński space is the simplest non-discrete topological space. It has important relations to the
theory of computation In theoretical computer science and mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation, using an algorithm, how algorithmic efficiency, efficiently they can be solved or t ...
and semantics. * If Γ is an
ordinal number In set theory illustrating the intersection of two sets Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a s ...
, then the set Γ = , Γ)_may_be_endowed_with_the_order_topology_generated_by_the_intervals_(''a'', ''b''),_[0, ''b'')_and_(''a'', Γ)_where_''a''_and_''b''_are_elements_of_Γ.

# _Continuous_functions

Continuity_is_expressed_in_terms_of_
, Γ)_may_be_endowed_with_the_order_topology_generated_by_the_intervals_(''a'', ''b''),_[0, ''b'')_and_(''a'', Γ)_where_''a''_and_''b''_are_elements_of_Γ.

# _Continuous_functions

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# _Continuous_functions

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# Continuous functions

Continuity is expressed in terms of neighborhood (topology)">neighborhood A neighbourhood (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...
s: is continuous at some point if and only if for any neighborhood of , there is a neighborhood of such that . Intuitively, continuity means no matter how "small" becomes, there is always a containing that maps inside and whose image under contains . This is equivalent to the condition that the Image (mathematics)#Inverse image">preimages
of the open (closed) sets in are open (closed) in . In metric spaces, this definition is equivalent to the epsilon-delta definition">ε–δ-definition that is often used in analysis. An extreme example: if a set is given the discrete topology, all functions :$f\colon X \rightarrow T$ to any topological space are continuous. On the other hand, if is equipped with the indiscrete topology and the space set is at least T0 space, T0, then the only continuous functions are the constant functions. Conversely, any function whose range is indiscrete is continuous.

## Alternative definitions

Several Characterizations of the category of topological spaces, equivalent definitions for a topological structure exist and thus there are several equivalent ways to define a continuous function.

### Neighborhood definition

Definitions based on preimages are often difficult to use directly. The following criterion expresses continuity in terms of
neighborhood A neighbourhood (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...
s: ''f'' is continuous at some point ''x'' ∈ ''X'' if and only if for any neighborhood ''V'' of ''f''(''x''), there is a neighborhood ''U'' of ''x'' such that ''f''(''U'') ⊆ ''V''. Intuitively, continuity means no matter how "small" ''V'' becomes, there is always a ''U'' containing ''x'' that maps inside ''V''. If ''X'' and ''Y'' are metric spaces, it is equivalent to consider the
neighborhood systemIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
of
open ball In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s centered at ''x'' and ''f''(''x'') instead of all neighborhoods. This gives back the above δ-ε definition of continuity in the context of metric spaces. However, in general topological spaces, there is no notion of nearness or distance. Note, however, that if the target space is , it is still true that ''f'' is continuous at ''a'' if and only if the limit of ''f'' as ''x'' approaches ''a'' is ''f''(''a''). At an isolated point, every function is continuous.

### Sequences and nets

In several contexts, the topology of a space is conveniently specified in terms of
limit points In mathematics, a limit point (or cluster point or accumulation point) of a Set (mathematics), set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every Neighbourhood (mathematics), neighbourhood ...
. In many instances, this is accomplished by specifying when a point is the
limit of a sequence As the positive integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. ...
, but for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points indexed by a
directed set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
, known as nets. A function is continuous only if it takes limits of sequences to limits of sequences. In the former case, preservation of limits is also sufficient; in the latter, a function may preserve all limits of sequences yet still fail to be continuous, and preservation of nets is a necessary and sufficient condition. In detail, a function ''f'': ''X'' → ''Y'' is sequentially continuous if whenever a sequence (''x''''n'') in ''X'' converges to a limit ''x'', the sequence (''f''(''x''''n'')) converges to ''f''(''x''). Thus sequentially continuous functions "preserve sequential limits". Every continuous function is sequentially continuous. If ''X'' is a
first-countable space In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...
and holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if ''X'' is a metric space, sequential continuity and continuity are equivalent. For non first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called
sequential space In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struc ...
s.) This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functions preserve limits of nets, and in fact this property characterizes continuous functions.

### Closure operator definition

Instead of specifying the open subsets of a topological space, the topology can also be determined by a
closure operatorIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
(denoted cl), which assigns to any subset ''A'' ⊆ ''X'' its closure, or an
interior operatorIn mathematics, a closure operator on a Set (mathematics), set ''S'' is a Function (mathematics), function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X, ...
(denoted int), which assigns to any subset ''A'' of ''X'' its
interior Interior may refer to: Arts and media * Interior (Degas), ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * Interior (play), ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * The Interior (novel) ...
. In these terms, a function :$f\colon \left(X,\mathrm\right) \to \left(X\text{'} ,\mathrm\text{'}\right)\,$ between topological spaces is continuous in the sense above if and only if for all subsets ''A'' of ''X'' :$f\left(\mathrm\left(A\right)\right) \subseteq \mathrm\text{'}\left(f\left(A\right)\right).$ That is to say, given any element ''x'' of ''X'' that is in the closure of any subset ''A'', ''f''(''x'') belongs to the closure of ''f''(''A''). This is equivalent to the requirement that for all subsets ''A''' of ''X''' :$f^\left(\mathrm\text{'}\left(A\text{'}\right)\right) \supseteq \mathrm\left(f^\left(A\text{'}\right)\right).$ Moreover, :$f\colon \left(X,\mathrm\right) \to \left(X\text{'} ,\mathrm\text{'}\right) \,$ is continuous if and only if :$f^\left(\mathrm\text{'}\left(A\right)\right) \subseteq \mathrm\left(f^\left(A\right)\right)$ for any subset ''A'' of ''X''.

## Properties

If ''f'': ''X'' → ''Y'' and ''g'': ''Y'' → ''Z'' are continuous, then so is the composition ''g'' ∘ ''f'': ''X'' → ''Z''. If ''f'': ''X'' → ''Y'' is continuous and * ''X'' is
compact Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
, then ''f''(''X'') is compact. * ''X'' is connected, then ''f''(''X'') is connected. * ''X'' is
path-connected In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...
, then ''f''(''X'') is path-connected. * ''X'' is Lindelöf, then ''f''(''X'') is Lindelöf. * ''X'' is separable, then ''f''(''X'') is separable. The possible topologies on a fixed set ''X'' are
partially ordered 250px, The set of all subsets of a three-element set , ordered by inclusion. Distinct sets on the same horizontal level are incomparable with each other. Some other pairs, such as and , are also incomparable. In mathematics, especially order the ...
: a topology τ1 is said to be coarser than another topology τ2 (notation: τ1 ⊆ τ2) if every open subset with respect to τ1 is also open with respect to τ2. Then, the :idX: (''X'', τ2) → (''X'', τ1) is continuous if and only if τ1 ⊆ τ2 (see also
comparison of topologiesIn topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as the c ...
). More generally, a continuous function :$\left(X, \tau_X\right) \rightarrow \left(Y, \tau_Y\right)$ stays continuous if the topology τ''Y'' is replaced by a coarser topology and/or τ''X'' is replaced by a
finer topologyIn topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as the c ...
.

## Homeomorphisms

Symmetric to the concept of a continuous map is an
open map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
, for which ''images'' of open sets are open. In fact, if an open map ''f'' has an
inverse function In mathematics, the inverse function of a Function (mathematics), function (also called the inverse of ) is a function (mathematics), function that undoes the operation of . The inverse of exists if and only if is Bijection, bijective, and i ...
, that inverse is continuous, and if a continuous map ''g'' has an inverse, that inverse is open. Given a
bijective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
function ''f'' between two topological spaces, the inverse function ''f''−1 need not be continuous. A bijective continuous function with continuous inverse function is called a ''
homeomorphism In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantiti ...
''. If a continuous bijection has as its
domain Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
a
compact space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
and its
codomain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

is , then it is a homeomorphism.

## Defining topologies via continuous functions

Given a function :$f\colon X \rightarrow S, \,$ where ''X'' is a topological space and ''S'' is a set (without a specified topology), the
final topology In general topology and related areas of mathematics, the final topology (or coinduced, strong, colimit, or inductive topology) on a Set (mathematics), set X, with respect to a family of functions from Topological space, topological spaces into X, ...
on ''S'' is defined by letting the open sets of ''S'' be those subsets ''A'' of ''S'' for which ''f''−1(''A'') is open in ''X''. If ''S'' has an existing topology, ''f'' is continuous with respect to this topology if and only if the existing topology is coarser than the final topology on ''S''. Thus the final topology can be characterized as the finest topology on ''S'' that makes ''f'' continuous. If ''f'' is
surjective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, this topology is canonically identified with the
quotient topology as the quotient space of a disk Disk or disc may refer to: * Disk (mathematics) * Disk storage Music * Disc (band), an American experimental music band * Disk (album), ''Disk'' (album), a 1995 EP by Moby Other uses * Disc (galaxy), a disc-sha ...
under the
equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
defined by ''f''. Dually, for a function ''f'' from a set ''S'' to a topological space, the
initial topology In general topology , a useful example in point-set topology. It is connected but not path-connected. In mathematics, general topology is the branch of topology that deals with the basic Set theory, set-theoretic definitions and constructions used ...
on ''S'' has as open subsets ''A'' of ''S'' those subsets for which ''f''(''A'') is open in ''X''. If ''S'' has an existing topology, ''f'' is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on ''S''. Thus the initial topology can be characterized as the coarsest topology on ''S'' that makes ''f'' continuous. If ''f'' is injective, this topology is canonically identified with the
subspace topologyIn topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a Topological_space#Definitions, topology induced from that of ''X'' called the subspace topology (or the relative ...

of ''S'', viewed as a subset of ''X''. A topology on a set ''S'' is uniquely determined by the class of all continuous functions $S \rightarrow X$ into all topological spaces ''X''. Dually, a similar idea can be applied to maps $X \rightarrow S.$

# Compact sets

Formally, a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
''X'' is called ''compact'' if each of its
open cover In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s has a
finite Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
subcover. Otherwise it is called ''non-compact''. Explicitly, this means that for every arbitrary collection :$\_$ of open subsets of such that :$X = \bigcup_ U_\alpha,$ there is a finite subset of such that :$X = \bigcup_ U_i.$ Some branches of mathematics such as
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

, typically influenced by the French school of Bourbaki, use the term ''quasi-compact'' for the general notion, and reserve the term ''compact'' for topological spaces that are both Hausdorff and ''quasi-compact''. A compact set is sometimes referred to as a ''compactum'', plural ''compacta''. Every closed interval in R of finite length is
compact Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
. More is true: In Rn, a set is compact
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
it is closed and bounded. (See
Heine–Borel theorem In real analysis the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states: For a subset ''S'' of Euclidean space R''n'', the following two statements are equivalent: *''S'' is closed set, closed and bounded set, bounded *''S'' i ...
). Every continuous image of a compact space is compact. A compact subset of a Hausdorff space is closed. Every continuous
bijection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

from a compact space to a Hausdorff space is necessarily a
homeomorphism In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantiti ...
. Every
sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

of points in a compact metric space has a convergent subsequence. Every compact finite-dimensional
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

can be embedded in some Euclidean space Rn.

# Connected sets

A
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
''X'' is said to be disconnected if it is the union of two
nonempty In mathematics, the empty set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exists by includ ...
open set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s. Otherwise, ''X'' is said to be connected. A
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of a topological space is said to be connected if it is connected under its
subspace topologyIn topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a Topological_space#Definitions, topology induced from that of ''X'' called the subspace topology (or the relative ...
. Some authors exclude the
empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, cha ...

(with its unique topology) as a connected space, but this article does not follow that practice. For a topological space ''X'' the following conditions are equivalent: #''X'' is connected. #''X'' cannot be divided into two disjoint nonempty
closed set In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
s. #The only subsets of ''X'' that are both open and closed (
clopen set upright=1.3, A Graph (discrete mathematics), graph with several clopen sets. Each of the three large pieces (i.e. connected component (topology), components) is a clopen set, as is the union of any two or all three. In topology, a clopen set (a po ...
s) are ''X'' and the empty set. #The only subsets of ''X'' with empty
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment *Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
are ''X'' and the empty set. #''X'' cannot be written as the union of two nonempty
separated sets In topology and related branches of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (ma ...
. #The only continuous functions from ''X'' to , the two-point space endowed with the discrete topology, are constant. Every interval in R is connected. The continuous image of a connected space is connected.

## Connected components

The maximal connected subsets (ordered by
inclusion Inclusion or Include may refer to: Sociology * Social inclusion, affirmative action to change the circumstances and habits that leads to social exclusion ** Inclusion (disability rights), including people with and without disabilities, people of ...

) of a nonempty topological space are called the connected components of the space. The components of any topological space ''X'' form a of ''X'': they are , nonempty, and their union is the whole space. Every component is a
closed subset In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
of the original space. It follows that, in the case where their number is finite, each component is also an open subset. However, if their number is infinite, this might not be the case; for instance, the connected components of the set of the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s are the one-point sets, which are not open. Let $\Gamma_x$ be the connected component of ''x'' in a topological space ''X'', and $\Gamma_x\text{'}$ be the intersection of all open-closed sets containing ''x'' (called quasi-component of ''x''.) Then $\Gamma_x \subset \Gamma\text{'}_x$ where the equality holds if ''X'' is compact Hausdorff or locally connected.

## Disconnected spaces

A space in which all components are one-point sets is called
totally disconnectedIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
. Related to this property, a space ''X'' is called totally separated if, for any two distinct elements ''x'' and ''y'' of ''X'', there exist disjoint open neighborhoods ''U'' of ''x'' and ''V'' of ''y'' such that ''X'' is the union of ''U'' and ''V''. Clearly any totally separated space is totally disconnected, but the converse does not hold. For example, take two copies of the rational numbers Q, and identify them at every point except zero. The resulting space, with the quotient topology, is totally disconnected. However, by considering the two copies of zero, one sees that the space is not totally separated. In fact, it is not even , and the condition of being totally separated is strictly stronger than the condition of being Hausdorff.

## Path-connected sets

A path from a point ''x'' to a point ''y'' in a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
''X'' is a
continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
''f'' from the
unit interval In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
,1to ''X'' with ''f''(0) = ''x'' and ''f''(1) = ''y''. A path-component of ''X'' is an
equivalence class In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of ''X'' under the
equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, which makes ''x'' equivalent to ''y'' if there is a path from ''x'' to ''y''. The space ''X'' is said to be path-connected (or pathwise connected or 0-connected) if there is at most one path-component, i.e. if there is a path joining any two points in ''X''. Again, many authors exclude the empty space. Every path-connected space is connected. The converse is not always true: examples of connected spaces that are not path-connected include the extended
long lineLong line or longline may refer to: *''Long Line'', an album by Peter Wolf *Long line (topology), or Alexandroff line, a topological space *Long line (telecommunications), a transmission line in a long-distance communications network *Longline fishi ...
''L''* and the ''
topologist's sine curve In the branch of mathematics known as topology, the topologist's sine curve or Warsaw sine curve is a topological space with several interesting properties that make it an important textbook example. It can be defined as the graph of a function, ...

''. However, subsets of the
real line In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
R are connected
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
they are path-connected; these subsets are the intervals of R. Also,
open subset Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australian ...
s of R''n'' or C''n'' are connected if and only if they are path-connected. Additionally, connectedness and path-connectedness are the same for
finite topological space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s.

# Products of spaces

Given ''X'' such that :$X := \prod_ X_i,$ is the Cartesian product of the topological spaces ''Xi'',
indexed Index may refer to: Arts, entertainment, and media Fictional entities * Index (A Certain Magical Index), Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo (megastr ...
by $i \in I$, and the canonical projections ''pi'' : ''X'' → ''Xi'', the product topology on ''X'' is defined as the
coarsest topologyIn topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as the c ...
(i.e. the topology with the fewest open sets) for which all the projections ''pi'' are
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
. The product topology is sometimes called the Tychonoff topology. The open sets in the product topology are unions (finite or infinite) of sets of the form $\prod_ U_i$, where each ''Ui'' is open in ''Xi'' and ''U''''i'' ≠ ''X''''i'' only finitely many times. In particular, for a finite product (in particular, for the product of two topological spaces), the products of base elements of the ''Xi'' gives a basis for the product $\prod_ X_i$. The product topology on ''X'' is the topology generated by sets of the form ''pi''−1(''U''), where ''i'' is in ''I '' and ''U'' is an open subset of ''Xi''. In other words, the sets form a
subbase In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...
for the topology on ''X''. A
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of ''X'' is open if and only if it is a (possibly infinite) union of intersections of finitely many sets of the form ''pi''−1(''U''). The ''pi''−1(''U'') are sometimes called
open cylinderIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s, and their intersections are cylinder sets. In general, the product of the topologies of each ''Xi'' forms a basis for what is called the box topology on ''X''. In general, the box topology is finer topology, finer than the product topology, but for finite products they coincide. Related to compactness is Tychonoff's theorem: the (arbitrary) product topology, product of compact spaces is compact.

# Separation axioms

Many of these names have alternative meanings in some of mathematical literature, as explained on History of the separation axioms; for example, the meanings of "normal" and "T4" are sometimes interchanged, similarly "regular" and "T3", etc. Many of the concepts also have several names; however, the one listed first is always least likely to be ambiguous. Most of these axioms have alternative definitions with the same meaning; the definitions given here fall into a consistent pattern that relates the various notions of separation defined in the previous section. Other possible definitions can be found in the individual articles. In all of the following definitions, ''X'' is again a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
. * ''X'' is ''T0 space, T0'', or ''Kolmogorov'', if any two distinct points in ''X'' are topological distinguishability, topologically distinguishable. (It is a common theme among the separation axioms to have one version of an axiom that requires T0 and one version that doesn't.) * ''X'' is '' T1'', or ''accessible'' or ''Fréchet'', if any two distinct points in ''X'' are separated. Thus, ''X'' is T1 if and only if it is both T0 and R0. (Though you may say such things as ''T1 space'', ''Fréchet topology'', and ''Suppose that the topological space ''X'' is Fréchet'', avoid saying ''Fréchet space'' in this context, since there is another entirely different notion of Fréchet space in
functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
.) * ''X'' is '''', or ''T2'' or ''separated'', if any two distinct points in ''X'' are separated by neighbourhoods. Thus, ''X'' is Hausdorff if and only if it is both T0 and R1. A Hausdorff space must also be T1. * ''X'' is ''Urysohn space, T'', or ''Urysohn'', if any two distinct points in ''X'' are separated by closed neighbourhoods. A T space must also be Hausdorff. * ''X'' is ''regular space, regular'', or ''T3'', if it is T0 and if given any point ''x'' and closed set ''F'' in ''X'' such that ''x'' does not belong to ''F'', they are separated by neighbourhoods. (In fact, in a regular space, any such ''x'' and ''F'' is also separated by closed neighbourhoods.) * ''X'' is ''Tychonoff space, Tychonoff'', or ''T'', ''completely T3'', or ''completely regular'', if it is T0 and if f, given any point ''x'' and closed set ''F'' in ''X'' such that ''x'' does not belong to ''F'', they are separated by a continuous function. * ''X'' is ''normal space, normal'', or ''T4'', if it is Hausdorff and if any two disjoint closed subsets of ''X'' are separated by neighbourhoods. (In fact, a space is normal if and only if any two disjoint closed sets can be separated by a continuous function; this is Urysohn's lemma.) * ''X'' is ''completely normal space, completely normal'', or ''T5'' or ''completely T4'', if it is T1 and if any two separated sets are separated by neighbourhoods. A completely normal space must also be normal. * ''X'' is ''perfectly normal space, perfectly normal'', or ''T6'' or ''perfectly T4'', if it is T1 and if any two disjoint closed sets are precisely separated by a continuous function. A perfectly normal Hausdorff space must also be completely normal Hausdorff. The Tietze extension theorem: In a normal space, every continuous real-valued function defined on a closed subspace can be extended to a continuous map defined on the whole space.

# Countability axioms

An axiom of countability is a property of certain mathematical objects (usually in a Category (mathematics), category) that requires the existence of a countable, countable set with certain properties, while without it such sets might not exist. Important countability axioms for
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
s: *
sequential space In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struc ...
: a set is open if every
sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

limit of a sequence, convergent to a point (geometry), point in the set is eventually in the set *
first-countable space In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...
: every point has a countable neighbourhood system, neighbourhood basis (local base) *second-countable space: the topology has a countable
base Base or BASE may refer to: Brands and enterprises * Base (mobile telephony provider), a Belgian mobile telecommunications operator *Base CRM Base CRM (originally Future Simple or PipeJump) is an enterprise software company based in Mountain Vie ...
*separable space: there exists a countable dense (topology), dense subspace *Lindelöf space: every
open cover In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
has a countable subcover *σ-compact space: there exists a countable cover by compact spaces Relations: *Every first countable space is sequential. *Every second-countable space is first-countable, separable, and Lindelöf. *Every σ-compact space is Lindelöf. *A
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
is first-countable. *For metric spaces second-countability, separability, and the Lindelöf property are all equivalent.

# Metric spaces

A metric space is an ordered pair $\left(M,d\right)$ where $M$ is a set and $d$ is a
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...
on $M$, i.e., a Function (mathematics), function :$d \colon M \times M \rightarrow \mathbb$ such that for any $x, y, z \in M$, the following holds: # $d\left(x,y\right) \ge 0$     (''non-negative''), # $d\left(x,y\right) = 0\,$ if and only if, iff $x = y\,$     (''identity of indiscernibles''), # $d\left(x,y\right) = d\left(y,x\right)\,$     (''symmetry'') and # $d\left(x,z\right) \le d\left(x,y\right) + d\left(y,z\right)$     (''triangle inequality'') . The function $d$ is also called ''distance function'' or simply ''distance''. Often, $d$ is omitted and one just writes $M$ for a metric space if it is clear from the context what metric is used. Every
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
is paracompact and , and thus normal space, normal. The metrization theorems provide necessary and sufficient conditions for a topology to come from a metric.

# Baire category theorem

The Baire category theorem says: If ''X'' is a completeness (topology), complete metric space or a locally compact Hausdorff space, then the interior of every union of countable, countably many nowhere dense sets is empty. Any open subspace of a Baire space is itself a Baire space.

# Main areas of research

## Continuum theory

A continuum (pl ''continua'') is a nonempty
compact Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
connected
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
, or less frequently, a
compact Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
connected
Hausdorff space In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...

. Continuum theory is the branch of topology devoted to the study of continua. These objects arise frequently in nearly all areas of topology and mathematical analysis, analysis, and their properties are strong enough to yield many 'geometric' features.

## Dynamical systems

Topological dynamics concerns the behavior of a space and its subspaces over time when subjected to continuous change. Many examples with applications to physics and other areas of math include fluid dynamics, dynamical billiards, billiards and geometric flow, flows on manifolds. The topological characteristics of fractals in fractal geometry, of Julia sets and the Mandelbrot set arising in complex dynamics, and of attractors in differential equations are often critical to understanding these systems.

## Pointless topology

Pointless topology (also called point-free or pointfree topology) is an approach to
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

that avoids mentioning points. The name 'pointless topology' is due to John von Neumann.Garrett Birkhoff, ''VON NEUMANN AND LATTICE THEORY'', ''John Von Neumann 1903-1957'', J. C. Oxtoley, B. J. Pettis, American Mathematical Soc., 1958, page 50-5 The ideas of pointless topology are closely related to mereotopology, mereotopologies, in which regions (sets) are treated as foundational without explicit reference to underlying point sets.

## Dimension theory

Dimension theory is a branch of general topology dealing with dimensional invariants of
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
s.

## Topological algebras

A topological algebra ''A'' over a topological field K is a topological vector space together with a continuous multiplication :$\cdot :A\times A \longrightarrow A$ :$\left(a,b\right)\longmapsto a\cdot b$ that makes it an algebra over a field, algebra over K. A unital associative algebra, associative topological algebra is a topological ring. The term was coined by David van Dantzig; it appears in the title of his Thesis, doctoral dissertation (1931).

## Metrizability theory

In
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

and related areas of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a metrizable space is a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
that is homeomorphism, homeomorphic to a
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
. That is, a topological space $\left(X,\tau\right)$ is said to be metrizable if there is a metric :$d\colon X \times X \to \left[0,\infty\right)$ such that the topology induced by ''d'' is $\tau$. Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable.

## Set-theoretic topology

Set-theoretic topology is a subject that combines set theory and general topology. It focuses on topological questions that are independent of Zermelo–Fraenkel set theory (ZFC). A famous problem is Moore space (topology)#Normal Moore space conjecture, the normal Moore space question, a question in general topology that was the subject of intense research. The answer to the normal Moore space question was eventually proved to be independent of ZFC.

*List of examples in general topology *Glossary of general topology for detailed definitions *List of general topology topics for related articles *Category of topological spaces