topology (structure)

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In , a topological space is, roughly speaking, a in which is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a of s, along with a set of s for each point, satisfying a set of s relating points and neighbourhoods. A topological space is the most general type of a that allows for the definition of , , and . Other spaces, such as s, s and s, are topological spaces with extra , properties or constraints. Although very general, topological spaces are a fundamental concept used in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called or .

# History

Around , discovered the $V - E + F = 2$ relating the number of vertices, edges and faces of a , and hence of a . The study and generalization of this formula, specifically by and , is at the origin of . In , published ''General investigations of curved surfaces'' which in section 3 defines the curved surface in a similar manner to the modern topological understanding: "A curved surface is said to possess continuous curvature at one of its points A, if the direction of all the straight lines drawn from A to points of the surface at an infinitely small distance from A are deflected infinitely little from one and the same plane passing through A." Yet, "until ’s work in the early 1850s, surfaces were always dealt with from a local point of view (as parametric surfaces) and topological issues were never considered." " and seem to be the first to realize that the main problem about the topology of (compact) surfaces is to find invariants (preferably numerical) to decide the equivalence of surfaces, that is, to decide whether two surfaces are homeomorphic or not." The subject is clearly defined by in his "" (1872): the geometry invariants of arbitrary continuous transformation, a kind of geometry. The term "topology" was introduced by in 1847, although he had used the term in correspondence some years earlier instead of previously used "Analysis situs". The foundation of this science, for a space of any dimension, was created by . His first article on this topic appeared in . In the 1930s, and first expressed the idea that a surface is a topological space that is . Topological spaces were first defined by in 1914 in his seminal "Principles of Set Theory". had been defined earlier in 1906 by , though it was Hausdorff who introduced the term "metric space".

# Definitions

The utility of the notion of a topology is shown by the fact that there are several equivalent definitions of this structure. Thus one chooses the suited for the application. The most commonly used is that in terms of , but perhaps more intuitive is that in terms of and so this is given first.

## Definition via neighbourhoods

This axiomatization is due to . Let $X$ be a set; the elements of $X$ are usually called , though they can be any mathematical object. We allow $X$ to be empty. Let $\mathcal$ be a assigning to each $x$ (point) in $X$ a non-empty collection $\mathcal\left(x\right)$ of subsets of $X.$ The elements of $\mathcal\left(x\right)$ will be called of $x$ with respect to $\mathcal$ (or, simply, ). The function $\mathcal$ is called a if the s below are satisfied; and then $X$ with $\mathcal$ is called a topological space. # If $N$ is a neighbourhood of $x$ (i.e., $N \in \mathcal\left(x\right)$), then $x \in N.$ In other words, each point belongs to every one of its neighbourhoods. # If $N$ is a subset of $X$ and includes a neighbourhood of $x,$ then $N$ is a neighbourhood of $x.$ I.e., every of a neighbourhood of a point $x \in X$ is again a neighbourhood of $x.$ # The of two neighbourhoods of $x$ is a neighbourhood of $x.$ # Any neighbourhood $N$ of $x$ includes a neighbourhood $M$ of $x$ such that $N$ is a neighbourhood of each point of $M.$ The first three axioms for neighbourhoods have a clear meaning. The fourth axiom has a very important use in the structure of the theory, that of linking together the neighbourhoods of different points of $X.$ A standard example of such a system of neighbourhoods is for the real line $\R,$ where a subset $N$ of $\R$ is defined to be a of a real number $x$ if it includes an open interval containing $x.$ Given such a structure, a subset $U$ of $X$ is defined to be open if $U$ is a neighbourhood of all points in $U.$ The open sets then satisfy the axioms given below. Conversely, when given the open sets of a topological space, the neighbourhoods satisfying the above axioms can be recovered by defining $N$ to be a neighbourhood of $x$ if $N$ includes an open set $U$ such that $x \in U.$

## Definition via open sets

A is an ordered pair $\left(X, \tau\right),$ where $X$ is a and $\tau$ is a collection of s of $X,$ satisfying the following s: # The and $X$ itself belong to $\tau.$ # Any arbitrary (finite or infinite) of members of $\tau$ belongs to $\tau.$ # The intersection of any finite number of members of $\tau$ belongs to $\tau.$ The elements of $\tau$ are called open sets and the collection $\tau$ is called a topology on $X.$ A subset $C \subseteq X$ is said to be in $\left(X, \tau\right)$ if and only if its $X \setminus C$ is an element of $\tau.$

### Examples of topologies

# Given $X = \,$ the or topology on $X$ is the $\tau = \ = \$ consisting of only the two subsets of $X$ required by the axioms forms a topology of $X.$ # Given $X = \,$ the family $\tau = \ = \$ of six subsets of $X$ forms another topology of $X.$ # Given $X = \,$ the on $X$ is the of $X,$ which is the family $\tau = \wp\left(X\right)$ consisting of all possible subsets of $X.$ In this case the topological space $\left(X, \tau\right)$ is called a . # Given $X = \Z,$ the set of integers, the family $\tau$ of all finite subsets of the integers plus $\Z$ itself is a topology, because (for example) the union of all finite sets not containing zero is not finite but is also not all of $\Z,$ and so it cannot be in $\tau.$

## Definition via closed sets

Using , the above axioms defining open sets become axioms defining s: # The empty set and $X$ are closed. # The intersection of any collection of closed sets is also closed. # The union of any finite number of closed sets is also closed. Using these axioms, another way to define a topological space is as a set $X$ together with a collection $\tau$ of closed subsets of $X.$ Thus the sets in the topology $\tau$ are the closed sets, and their complements in $X$ are the open sets.

## Other definitions

There are many other equivalent ways to define a topological space: in other words the concepts of neighbourhood, or that of open or closed sets can be reconstructed from other starting points and satisfy the correct axioms. Another way to define a topological space is by using the , which define the closed sets as the of an on the of $X.$ A is a generalisation of the concept of . A topology is completely determined if for every net in $X$ the set of its is specified.

# Comparison of topologies

A variety of topologies can be placed on a set to form a topological space. When every set in a topology $\tau_1$ is also in a topology $\tau_2$ and $\tau_1$ is a subset of $\tau_2,$ we say that $\tau_2$is than $\tau_1,$ and $\tau_1$ is than $\tau_2.$ A proof that relies only on the existence of certain open sets will also hold for any finer topology, and similarly a proof that relies only on certain sets not being open applies to any coarser topology. The terms and are sometimes used in place of finer and coarser, respectively. The terms and are also used in the literature, but with little agreement on the meaning, so one should always be sure of an author's convention when reading. The collection of all topologies on a given fixed set $X$ forms a : if $F = \left\$ is a collection of topologies on $X,$ then the of $F$ is the intersection of $F,$ and the of $F$ is the meet of the collection of all topologies on $X$ that contain every member of $F.$

# Continuous functions

A $f : X \to Y$ between topological spaces is called if for every $x \in X$ and every neighbourhood $N$ of $f\left(x\right)$ there is a neighbourhood $M$ of $x$ such that $f\left(M\right) \subseteq N.$ This relates easily to the usual definition in analysis. Equivalently, $f$ is continuous if the of every open set is open. This is an attempt to capture the intuition that there are no "jumps" or "separations" in the function. A is a that is continuous and whose is also continuous. Two spaces are called if there exists a homeomorphism between them. From the standpoint of topology, homeomorphic spaces are essentially identical. In , one of the fundamental is Top, which denotes the whose are topological spaces and whose s are continuous functions. The attempt to classify the objects of this category ( ) by s has motivated areas of research, such as , , and .

# 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. Any set can be given the 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 (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 s where limit points are unique.

## Metric spaces

Metric spaces embody a , a precise notion of distance between points. Every 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 . On a finite-dimensional this topology is the same for all norms. There are many ways of defining a topology on $\R,$ the set of s. The standard topology on $\R$ is generated by the . The set of all open intervals forms a 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 s $\R^n$ can be given a topology. In the usual topology on $\R^n$ the basic open sets are the open s. Similarly, $\C,$ the set of s, and $\C^n$ have a standard topology in which the basic open sets are open balls.

## Proximity spaces

s provide a notion of closeness of two sets.

## Uniform spaces

Uniform spaces axiomatize ordering the distance between distinct points.

## Function spaces

A topological space in which the are functions is called a .

## Cauchy spaces

s axiomatize the ability to test whether a net is . Cauchy spaces provide a general setting for studying s.

## Convergence spaces

s capture some of the features of convergence of .

## Grothendieck sites

s are with additional data axiomatizing whether a family of arrows covers an object. Sites are a general setting for defining .

## Other spaces

If $\Gamma$ is a on a set $X$ then $\ \cup \Gamma$ is a topology on $X.$ Many sets of s in are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function. Any has a topology native to it, and this can be extended to vector spaces over that field. Every has a since it is locally Euclidean. Similarly, every and every inherits a natural topology from . The is defined algebraically on the or an . On $\R^n$ or $\C^n,$ the closed sets of the Zariski topology are the s of systems of equations. A has a natural topology that generalizes many of the geometric aspects of s with and . The is the simplest non-discrete topological space. It has important relations to the and semantics. There exist numerous topologies on any given . Such spaces are called s. Finite spaces are sometimes used to provide examples or counterexamples to conjectures about topological spaces in general. Any set can be given the in which the open sets are the empty set and the sets whose complement is finite. This is the smallest topology on any infinite set. Any set can be given the , 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. The real line can also be given the . Here, the basic open sets are the half open intervals

# __Topological_constructions_

Every_subset_of_a_topological_space_can_be_given_the__in_which_the_open_sets_are_the_intersections_of_the_open_sets_of_the_larger_space_with_the_subset.__For_any__of_topological_spaces,_the_product_can_be_given_the_,_which_is_generated_by_the_inverse_images_of_open_sets_of_the_factors_under_the__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__is_defined_as_follows:_if_$X$_is_a_topological_space_and_$Y$_is_a_set,_and_if_$f_:_X_\to_Y$_is_a__,_then_the_quotient_topology_on_$Y$_is_the_collection_of_subsets_of_$Y$_that_have_open_s_under_$f.$_In_other_words,_the__is_the_finest_topology_on_$Y$_for_which_$f$_is_continuous.__A_common_example_of_a_quotient_topology_is_when_an__is_defined_on_the_topological_space_$X.$__The_map_$f$_is_then_the_natural_projection_onto_the_set_of_es. The_Vietoris_topology_on_the_set_of_all_non-empty_subsets_of_a_topological_space_$X,$_named_for_,_is_generated_by_the_following_basis:_for_every_$n$-tuple_$U_1,_\ldots,_U_n$_of_open_sets_in_$X,$_we_construct_a_basis_set_consisting_of_all_subsets_of_the_union_of_the_$U_i$_that_have_non-empty_intersections_with_each_$U_i.$_ The_Fell_topology_on_the_set_of_all_non-empty_closed_subsets_of_a___$X$_is_a_variant_of_the_Vietoris_topology,_and_is_named_after_mathematician_James_Fell._It_is_generated_by_the_following_basis:_for_every_$n$-tuple_$U_1,_\ldots,_U_n$_of_open_sets_in_$X$_and_for_every_compact_set_$K,$_the_set_of_all_subsets_of_$X$_that_are_disjoint_from_$K$_and_have_nonempty_intersections_with_each_$U_i$_is_a_member_of_the_basis.

# __Topological_spaces_with_order_structure_

*_Spectral.__A_space_is__if_and_only_if_it_is_the_prime__(_theorem). *_Specialization_preorder.__In_a_space_the__is_defined_by_$x_\leq_y$_if_and_only_if_$\operatorname\_\subseteq_\operatorname\,$_where_$\operatorname$_denotes_an_operator_satisfying_the__.

# __See_also_

*_ *__–_The_system_of_all_open_sets_of_a_given_topological_space_ordered_by_inclusion_is_a_complete_Heyting_algebra. *_ *_ *_ *_ *_ *_ *_ *_ *_

# __Bibliography_

*_ {{Authority_control Topological_spaces.html" ;"title=", 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. If $\Gamma$ is an , then the set $\Gamma = \left[0, \Gamma\right)$ may be endowed with the generated by the intervals $\left(a, b\right),$ $\left[0, b\right),$ and $\left(a, \Gamma\right)$ where $a$ and $b$ are elements of $\Gamma.$ of a $F_n$ consists of the so-called "marked metric graph structures" of volume 1 on $F_n.$

# Topological constructions

Every subset of a topological space can be given the in which the open sets are the intersections of the open sets of the larger space with the subset. For any of topological spaces, the product can be given the , which is generated by the inverse images of open sets of the factors under the 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 is defined as follows: if $X$ is a topological space and $Y$ is a set, and if $f : X \to Y$ is a , then the quotient topology on $Y$ is the collection of subsets of $Y$ that have open s under $f.$ In other words, the is the finest topology on $Y$ for which $f$ is continuous. A common example of a quotient topology is when an is defined on the topological space $X.$ The map $f$ is then the natural projection onto the set of es. The Vietoris topology on the set of all non-empty subsets of a topological space $X,$ named for , is generated by the following basis: for every $n$-tuple $U_1, \ldots, U_n$ of open sets in $X,$ we construct a basis set consisting of all subsets of the union of the $U_i$ that have non-empty intersections with each $U_i.$ The Fell topology on the set of all non-empty closed subsets of a $X$ is a variant of the Vietoris topology, and is named after mathematician James Fell. It is generated by the following basis: for every $n$-tuple $U_1, \ldots, U_n$ of open sets in $X$ and for every compact set $K,$ the set of all subsets of $X$ that are disjoint from $K$ and have nonempty intersections with each $U_i$ is a member of the basis.

# Classification of topological spaces

Topological spaces can be broadly classified, homeomorphism, by their . A topological property is a property of spaces that is invariant under homeomorphisms. To prove that two spaces are not homeomorphic it is sufficient to find a topological property not shared by them. Examples of such properties include , , and various s. For algebraic invariants see .

# Topological spaces with algebraic structure

For any we can introduce the discrete topology, under which the algebraic operations are continuous functions. For any such structure that is not finite, we often have a natural topology compatible with the algebraic operations, in the sense that the algebraic operations are still continuous. This leads to concepts such as s, s, s and s.

# Topological spaces with order structure

* Spectral. A space is if and only if it is the prime ( theorem). * Specialization preorder. In a space the is defined by $x \leq y$ if and only if $\operatorname\ \subseteq \operatorname\,$ where $\operatorname$ denotes an operator satisfying the .