Lila: Created page with "In mathematics, a '''topological space''' is, roughly speaking, a geometrical space in which ''closeness'' is defined but cannot n..."

2021-10-25T20:22:26Z

Created page with "In mathematics, a '''topological space''' is, roughly speaking, a geometrical space in which ''closeness'' is defined but cannot n..."

New page

In [[mathematics]], a '''topological space''' is, roughly speaking, a [[Geometry|geometrical space]] in which [[Closeness (mathematics)|''closeness'']] is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a [[Set (mathematics)|set]] of [[Point (geometry)|point]]s, along with a set of [[Neighbourhood (mathematics)|neighbourhood]]s for each point, satisfying a set of [[Axiom#Non-logical axioms|axiom]]s relating points and neighbourhoods.

A topological space is the most general type of a [[Space (mathematics)|mathematical space]] that allows for the definition of [[Limit (mathematics)|limits]], [[Continuous function (topology)|continuity]], and [[Connected space|connectedness]].<ref>{{harvnb|Schubert|1968|loc=p. 13}}</ref> Other spaces, such as [[Euclidean space]]s, [[metric space]]s and [[manifold]]s, are topological spaces with extra [[Mathematical structure|structures]], 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 [[point-set topology]] or [[general topology]].

== Definitions ==
{{main|Characterizations of the category of topological spaces}}

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 [[axiomatization]] suited for the application. The most commonly used is that in terms of {{em|[[open set]]s}}, but perhaps more intuitive is that in terms of {{em|[[Neighbourhood (mathematics)|neighbourhood]]s}} and so this is given first.

[[Image:Topological space examples.svg|frame|right|300px|Four examples and two non-examples of topologies on the three-point set <math>\{1,2,3\}.</math> The bottom-left example is not a topology because the union of <math>\{2\}</math> and <math>\{3\}</math> [i.e. <math>\{2,3\}</math>] is missing; the bottom-right example is not a topology because the intersection of <math>\{1,2\}</math> and <math>\{2,3\}</math> [i.e. <math>\{2\}</math>], is missing.]]

=== Definition via neighbourhoods{{anchor|Neighborhood definition|Neighbourhood definition}} ===

This axiomatization is due to [[Felix Hausdorff]].
Let <math>X</math> be a set; the elements of <math>X</math> are usually called {{em|points}}, though they can be any mathematical object. We allow <math>X</math> to be empty. Let <math>\mathcal{N}</math> be a [[Function (mathematics)|function]] assigning to each <math>x</math> (point) in <math>X</math> a non-empty collection <math>\mathcal{N}(x)</math> of subsets of <math>X.</math> The elements of <math>\mathcal{N}(x)</math> will be called {{em|neighbourhoods}} of <math>x</math> with respect to <math>\mathcal{N}</math> (or, simply, {{em|neighbourhoods of <math>x</math>}}). The function <math>\mathcal{N}</math> is called a [[Neighbourhood (topology)|neighbourhood topology]] if the [[axiom]]s below{{sfn|Brown|2006|loc=section 2.1}} are satisfied; and then <math>X</math> with <math>\mathcal{N}</math> is called a '''topological space'''.

# If <math>N</math> is a neighbourhood of <math>x</math> (i.e., <math>N \in \mathcal{N}(x)</math>), then <math>x \in N.</math> In other words, each point belongs to every one of its neighbourhoods.
# If <math>N</math> is a subset of <math>X</math> and includes a neighbourhood of <math>x,</math> then <math>N</math> is a neighbourhood of <math>x.</math> I.e., every [[superset]] of a neighbourhood of a point <math>x \in X</math> is again a neighbourhood of <math>x.</math>
# The [[Intersection (set theory)|intersection]] of two neighbourhoods of <math>x</math> is a neighbourhood of <math>x.</math>
# Any neighbourhood <math>N</math> of <math>x</math> includes a neighbourhood <math>M</math> of <math>x</math> such that <math>N</math> is a neighbourhood of each point of <math>M.</math>

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 <math>X.</math>

A standard example of such a system of neighbourhoods is for the real line <math>\R,</math> where a subset <math>N</math> of <math>\R</math> is defined to be a {{em|neighbourhood}} of a real number <math>x</math> if it includes an open interval containing <math>x.</math>

Given such a structure, a subset <math>U</math> of <math>X</math> is defined to be '''open''' if <math>U</math> is a neighbourhood of all points in <math>U.</math> 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 <math>N</math> to be a neighbourhood of <math>x</math> if <math>N</math> includes an open set <math>U</math> such that <math>x \in U.</math>{{sfn|Brown|2006|loc=section 2.2}}

=== Definition via open sets {{anchor|topology}} ===

A {{em|{{visible anchor|topological space}}}} is an ordered pair <math>(X, \tau),</math> where <math>X</math> is a [[Set (mathematics)|set]] and <math>\tau</math> is a collection of [[subset]]s of <math>X,</math> satisfying the following [[axiom]]s:{{sfn|Armstrong|1983|loc=definition 2.1}}

# The [[empty set]] and <math>X</math> itself belong to <math>\tau.</math>
# Any arbitrary (finite or infinite) [[Union (set theory)|union]] of members of <math>\tau</math> belongs to <math>\tau.</math>
# The intersection of any finite number of members of <math>\tau</math> belongs to <math>\tau.</math>

The elements of <math>\tau</math> are called '''open sets''' and the collection <math>\tau</math> is called a '''topology''' on <math>X.</math> A subset <math>C \subseteq X</math> is said to be {{em|closed}} in <math>(X, \tau)</math> if and only if its [[Complement (set theory)|complement]] <math>X \setminus C</math> is an element of <math>\tau.</math>

==== Examples of topologies ====
<ol>
<li>Given <math>X = \{ 1, 2, 3, 4\},</math> the [[Trivial topology|trivial]] or {{em|indiscrete}} topology on <math>X</math> is the [[Family of sets|family]] <math>\tau = \{ \{ \}, \{ 1, 2, 3, 4 \} \} = \{ \varnothing, X \}</math> consisting of only the two subsets of <math>X</math> required by the axioms forms a topology of <math>X.</math></li>
<li>Given <math>X = \{ 1, 2, 3, 4\},</math> the family
:<math>\tau = \{ \{ \}, \{ 2 \}, \{1, 2\}, \{2, 3\}, \{1, 2, 3\}, \{ 1, 2, 3, 4 \} \} = \{ \varnothing, \{ 2 \}, \{1, 2\}, \{2, 3\}, \{1, 2, 3\}, X \}</math>
of six subsets of <math>X</math> forms another topology of <math>X.</math></li>
<li>Given <math>X = \{ 1, 2, 3, 4\},</math> the [[discrete topology]] on <math>X</math> is the [[power set]] of <math>X,</math> which is the family <math>\tau = \wp(X)</math> consisting of all possible subsets of <math>X.</math> In this case the topological space <math>(X, \tau)</math> is called a {{em|discrete space}}.</li>
<li>Given <math>X = \Z,</math> the set of integers, the family <math>\tau</math> of all finite subsets of the integers plus <math>\Z</math> itself is {{em|not}} a topology, because (for example) the union of all finite sets not containing zero is not finite but is also not all of <math>\Z,</math> and so it cannot be in <math>\tau.</math></li>
</ol>

=== Definition via closed sets ===
Using [[de Morgan's laws]], the above axioms defining open sets become axioms defining '''[[closed set]]s''':

# The empty set and <math>X</math> 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 <math>X</math> together with a collection <math>\tau</math> of closed subsets of <math>X.</math> Thus the sets in the topology <math>\tau</math> are the closed sets, and their complements in <math>X</math> 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 [[Kuratowski closure axioms]], which define the closed sets as the [[Fixed point (mathematics)|fixed points]] of an [[Operator (mathematics)|operator]] on the [[power set]] of <math>X.</math>

A [[Net (mathematics)|net]] is a generalisation of the concept of [[sequence]]. A topology is completely determined if for every net in <math>X</math> the set of its [[Topology glossary|accumulation points]] is specified.

== Comparison of topologies ==
{{main|Comparison of topologies}}

A variety of topologies can be placed on a set to form a topological space. When every set in a topology <math>\tau_1</math> is also in a topology <math>\tau_2</math> and <math>\tau_1</math> is a subset of <math>\tau_2,</math> we say that <math>\tau_2</math>is {{em|finer}} than <math>\tau_1,</math> and <math>\tau_1</math> is {{em|coarser}} than <math>\tau_2.</math> 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 {{em|larger}} and {{em|smaller}} are sometimes used in place of finer and coarser, respectively. The terms {{em|stronger}} and {{em|weaker}} 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 <math>X</math> forms a [[complete lattice]]: if <math>F = \left\{ \tau_{\alpha} : \alpha \in A \right\}</math> is a collection of topologies on <math>X,</math> then the [[Infimum#Infima within partially ordered sets|meet]] of <math>F</math> is the intersection of <math>F,</math> and the [[Supremum#Suprema within partially ordered sets|join]] of <math>F</math> is the meet of the collection of all topologies on <math>X</math> that contain every member of <math>F.</math>

== Continuous functions ==
{{main|Continuous function}}

A [[Function (mathematics)|function]] <math>f : X \to Y</math> between topological spaces is called '''[[Continuity (topology)|continuous]]''' if for every and every neighbourhood <math>N</math> of <math>f(x)</math> there is a neighbourhood <math>M</math> of <math>x</math> such that <math>f(M) \subseteq N.</math> This relates easily to the usual definition in analysis. Equivalently, <math>f</math> is continuous if the [[inverse image]] of every open set is open.{{sfn|Armstrong|1983|loc=theorem 2.6}} This is an attempt to capture the intuition that there are no "jumps" or "separations" in the function. A [[homeomorphism]] is a [[bijection]] that is continuous and whose [[Inverse function|inverse]] is also continuous. Two spaces are called {{em|homeomorphic}} if there exists a homeomorphism between them. From the standpoint of topology, homeomorphic spaces are essentially identical.<ref>{{Cite book|isbn = 978-93-325-4953-1|last = Munkres|first = James R|title = Topology|date = 2015|pages = 317–319}}</ref>

In [[category theory]], one of the fundamental [[Category (mathematics)|categories]] is '''Top''', which denotes the [[category of topological spaces]] whose [[Object (category theory)|objects]] are topological spaces and whose [[morphism]]s are continuous functions. The attempt to classify the objects of this category ([[up to]] [[homeomorphism]]) by [[Invariant (mathematics)|invariant]]s has motivated areas of research, such as [[Homotopy|homotopy theory]], [[Homology (mathematics)|homology theory]], and [[K-theory]].

== 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 [[discrete space|discrete topology]] 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 topology]] (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]]s where limit points are unique.

=== Metric spaces ===
{{main|Metric space}}

Metric spaces embody a [[Metric (mathematics)|metric]], a precise notion of distance between points.

Every [[metric space]] 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.

There are many ways of defining a topology on <math>\R,</math> the set of [[real number]]s. The standard topology on <math>\R</math> is generated by the [[Interval (mathematics)#Terminology|open intervals]]. The set of all open intervals forms a [[base (topology)|base]] 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]]s <math>\R^n</math> can be given a topology. In the '''usual topology''' on <math>\R^n</math> the basic open sets are the open [[Ball (mathematics)|ball]]s. Similarly, <math>\C,</math> the set of [[complex number]]s, and <math>\C^n</math> have a standard topology in which the basic open sets are open balls.

=== Proximity spaces ===
{{main|proximity space}}

[[Proximity space]]s provide a notion of closeness of two sets.

{{expand section|date=November 2016}}

=== Uniform spaces ===
{{main|uniform space}}

Uniform spaces axiomatize ordering the distance between distinct points.

{{expand section|date=November 2016}}

=== Function spaces ===
{{main|function space}}

A topological space in which the {{em|points}} are functions is called a [[function space]].

{{expand section|date=November 2016}}

=== Cauchy spaces ===
{{main|Cauchy space}}

[[Cauchy space]]s axiomatize the ability to test whether a net is [[Cauchy net|Cauchy]]. Cauchy spaces provide a general setting for studying [[complete space|completion]]s.

{{expand section|date=November 2016}}

=== Convergence spaces ===
[[Convergence space]]s capture some of the features of convergence of [[Filter (mathematics)|filters]].

{{expand section|date=November 2016}}

=== Grothendieck sites ===
{{main|Grothendieck site}}

[[Grothendieck site]]s are [[Category (mathematics)|categories]] with additional data axiomatizing whether a family of arrows covers an object. Sites are a general setting for defining [[Sheaf (mathematics)|sheaves]].

{{expand section|date=November 2016}}

=== Other spaces ===

If <math>\Gamma</math> is a [[Filter (mathematics)|filter]] on a set <math>X</math> then <math>\{ \varnothing \} \cup \Gamma</math> is a topology on <math>X.</math>

Many sets of [[linear operator]]s in [[functional analysis]] are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function.

Any [[local field]] has a topology native to it, and this can be extended to vector spaces over that field.

Every [[manifold]] has a [[natural topology]] since it is locally Euclidean. Similarly, every [[simplex]] and every [[simplicial complex]] inherits a natural topology from .

The [[Zariski topology]] is defined algebraically on the [[spectrum of a ring]] or an [[algebraic variety]]. On <math>\R^n</math> or <math>\C^n,</math> the closed sets of the Zariski topology are the [[solution set]]s of systems of [[polynomial]] equations.

A [[linear graph]] has a natural topology that generalizes many of the geometric aspects of [[Graph theory|graph]]s with [[Vertex (graph theory)|vertices]] and [[Graph (discrete mathematics)#Graph|edges]].

The [[Sierpiński space]] is the simplest non-discrete topological space. It has important relations to the [[theory of computation]] and semantics.

There exist numerous topologies on any given [[finite set]]. Such spaces are called [[finite topological space]]s. Finite spaces are sometimes used to provide examples or counterexamples to conjectures about topological spaces in general.

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 space|T<sub>1</sub>]] topology on any infinite set.{{citation needed|date=June 2021}}

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.

The real line can also be given the [[lower limit topology]]. Here, the basic open sets are the half open intervals <math>[a, b).</math> This topology on <math>\R</math> 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 <math>\Gamma</math> is an [[ordinal number]], then the set <math>\Gamma = [0, \Gamma)</math> may be endowed with the [[order topology]] generated by the intervals <math>(a, b),</math> <math>[0, b),</math> and <math>(a, \Gamma)</math> where <math>a</math> and <math>b</math> are elements of <math>\Gamma.</math>

[[Outer space (mathematics)|Outer space]] of a [[free group]] <math>F_n</math> consists of the so-called "marked metric graph structures" of volume 1 on <math>F_n.</math><ref name="CV86">{{cite journal|last1= Culler|first1= Marc|author-link= Marc Culler|last2= Vogtmann|first2= Karen|author-link2= Karen Vogtmann|title= Moduli of graphs and automorphisms of free groups|journal=[[Inventiones Mathematicae]]|volume= 84|issue= 1|pages= 91–119|date= 1986|url= http://www.math.cornell.edu/~vogtmann/ScannedPapers/1986.0084.pdf|doi= 10.1007/BF01388734 |bibcode= 1986InMat..84...91C|s2cid= 122869546}}</ref>

== Topological constructions ==
Every subset of a topological space can be given the [[subspace topology]] in which the open sets are the intersections of the open sets of the larger space with the subset. For any [[indexed family]] of topological spaces, the product can be given the [[product topology]], which is generated by the inverse images of open sets of the factors under the [[Projection (mathematics)|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 (topology)|quotient space]] is defined as follows: if <math>X</math> is a topological space and <math>Y</math> is a set, and if <math>f : X \to Y</math> is a [[Surjection|surjective]] [[Function (mathematics)|function]], then the quotient topology on <math>Y</math> is the collection of subsets of <math>Y</math> that have open [[inverse image]]s under <math>f.</math> In other words, the [[quotient topology]] is the finest topology on <math>Y</math> for which <math>f</math> is continuous. A common example of a quotient topology is when an [[equivalence relation]] is defined on the topological space <math>X.</math> The map <math>f</math> is then the natural projection onto the set of [[equivalence class]]es.

The '''Vietoris topology''' on the set of all non-empty subsets of a topological space <math>X,</math> named for [[Leopold Vietoris]], is generated by the following basis: for every <math>n</math>-tuple <math>U_1, \ldots, U_n</math> of open sets in <math>X,</math> we construct a basis set consisting of all subsets of the union of the <math>U_i</math> that have non-empty intersections with each <math>U_i.</math>

The '''Fell topology''' on the set of all non-empty closed subsets of a [[locally compact]] [[Polish space]] <math>X</math> is a variant of the Vietoris topology, and is named after mathematician James Fell. It is generated by the following basis: for every <math>n</math>-tuple <math>U_1, \ldots, U_n</math> of open sets in <math>X</math> and for every compact set <math>K,</math> the set of all subsets of <math>X</math> that are disjoint from <math>K</math> and have nonempty intersections with each <math>U_i</math> is a member of the basis.

== Classification of topological spaces ==
{{main|Topological property}}
Topological spaces can be broadly classified, [[up to]] homeomorphism, by their [[topological properties]]. 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 [[Connectedness (topology)|connectedness]], [[Compactness (topology)|compactness]], and various [[separation axiom]]s. For algebraic invariants see [[algebraic topology]].

== Topological spaces with algebraic structure ==
For any [[Algebraic structure|algebraic objects]] 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 [[topological group]]s, [[topological vector space]]s, [[topological ring]]s and [[local field]]s.

== Topological spaces with order structure ==
* '''Spectral'''. A space is [[Spectral space|spectral]] if and only if it is the prime [[spectrum of a ring]] ([[Melvin Hochster|Hochster]] theorem).
* '''Specialization preorder'''. In a space the [[Specialization preorder|'''specialization''' (or '''canonical''') '''preorder''']] is defined by <math>x \leq y</math> if and only if <math>\operatorname{cl}\{ x \} \subseteq \operatorname{cl}\{ y \},</math> where <math>\operatorname{cl}</math> denotes an operator satisfying the [[Kuratowski closure axioms]].

==Licensing==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Topological_space Topological space, Wikipedia] under a CC BY-SA license

Topological Spaces - Revision history

Khanh at 15:58, 29 October 2021

Lila: Created page with "In mathematics, a '''topological space''' is, roughly speaking, a geometrical space in which ''closeness'' is defined but cannot n..."