In

_{τ}'' may be used to denote a set ''X'' endowed with the particular topology ''τ''.
The members of ''τ'' are called ''

_{1} 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.

^{''n''} can be given a topology. In the usual topology on R^{''n''} the basic open sets are the open ^{''n''} have a standard topology in which the basic open sets are open balls.
The real line can also be given the [ ''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.

^{n}.
* The ^{''n''} or C^{''n''}, the closed sets of the Zariski topology are the

_
A_neighbourhood_(British_English_
British_English_(BrE)_is_the_standard_dialect
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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 thereal 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 afamily
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 ''Xopen 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 atopological 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 thesubspace 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
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, 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 thediscrete 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 TTopologies on the real and complex numbers

There are many ways to define a topology on R, the set ofreal 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 Rball
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 Clower 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 The metric topology

Everymetric 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 calledfinite 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 RZariski 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 Rsolution 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

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_
A_neighbourhood_(British_English_
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_Continuous_functions

Continuity_is_expressed_in_terms_of_neighborhood_(topology)">neighborhood_Continuous_functions

Continuity_is_expressed_in_terms_of_neighborhood_(topology)">neighborhoodContinuous 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 :$f\backslash colon\; X\; \backslash rightarrow\; T$ 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, T

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 ofneighborhood
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 oflimit 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''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 aclosure 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
:$f\backslash colon\; (X,\backslash mathrm)\; \backslash to\; (X\text{'}\; ,\backslash mathrm\text{'})\backslash ,$
between topological spaces is continuous in the sense above if and only if for all subsets ''A'' of ''X''
:$f(\backslash mathrm(A))\; \backslash subseteq\; \backslash mathrm\text{'}(f(A)).$
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''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'' iscompact
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 τcomparison 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
:$(X,\; \backslash tau\_X)\; \backslash rightarrow\; (Y,\; \backslash tau\_Y)$
stays continuous if the topology τfiner 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 anopen 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''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 :$f\backslash colon\; X\; \backslash rightarrow\; S,\; \backslash ,$ where ''X'' is a topological space and ''S'' is a set (without a specified topology), thefinal 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''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 $S\; \backslash rightarrow\; X$ into all topological spaces ''X''. Dually, a similar idea can be applied to maps $X\; \backslash rightarrow\; S.$
Compact sets

Formally, atopological 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
:$\backslash \_$
of open subsets of such that
:$X\; =\; \backslash bigcup\_\; U\_\backslash alpha,$
there is a finite subset of such that
:$X\; =\; \backslash bigcup\_\; U\_i.$
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 Rif 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 RConnected sets

Anonempty
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 ...

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 byinclusion
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 $\backslash Gamma\_x$ be the connected component of ''x'' in a topological space ''X'', and $\backslash Gamma\_x\text{'}$ be the intersection of all open-closed sets containing ''x'' (called quasi-component of ''x''.) Then $\backslash Gamma\_x\; \backslash subset\; \backslash Gamma\text{'}\_x$ 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 calledtotally 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 acontinuous 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 Rfinite 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 :$X\; :=\; \backslash prod\_\; X\_i,$ is the Cartesian product of the topological spaces ''Xindexed
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 $i\; \backslash in\; I$, and the canonical projections ''pcoarsest 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 ''pcontinuous
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 $\backslash prod\_\; U\_i$, where each ''Usubbase
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 ''popen 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 ''XSeparation 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 "Tfunctional 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 ''TCountability 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 forsequential 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 $(M,d)$ where $M$ is a set and $d$ is ametric
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
:$d\; \backslash colon\; M\; \backslash times\; M\; \backslash rightarrow\; \backslash mathbb$
such that for any $x,\; y,\; z\; \backslash in\; M$, the following holds:
# $d(x,y)\; \backslash ge\; 0$ (''non-negative''),
# $d(x,y)\; =\; 0\backslash ,$ if and only if, iff $x\; =\; y\backslash ,$ (''identity of indiscernibles''),
# $d(x,y)\; =\; d(y,x)\backslash ,$ (''symmetry'') and
# $d(x,z)\; \backslash le\; d(x,y)\; +\; d(y,z)$ (''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 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 nonemptycompact
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 totopology
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 ofTopological algebras

A topological algebra ''A'' over a topological field K is a topological vector space together with a continuous multiplication :$\backslash cdot\; :A\backslash times\; A\; \backslash longrightarrow\; A$ :$(a,b)\backslash longmapsto\; a\backslash cdot\; b$ 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

Inmathematics
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 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.See also

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

Further reading

Some standard books on general topology include: * Bourbaki, Topologie Générale (General Topology), . * John L. Kelley (1955''General Topology''

link from Internet Archive, originally published by David Van Nostrand Company. * Stephen Willard, General Topology, . * James Munkres, Topology, . * George F. Simmons, Introduction to Topology and Modern Analysis, . * Paul L. Shick, Topology: Point-Set and Geometric, . * Ryszard Engelking, General Topology, . * * O.Ya. Viro, O.A. Ivanov, V.M. Kharlamov and N.Yu. Netsvetaev

Elementary Topology: Textbook in Problems

.

Topological Shapes and their Significance

by K.A.Rousan arvXiv id- 1905.13481 The arXiv subject code i

math.GN

External links

* {{Areas of mathematics , collapsed General topology,