Editing Topological space (section)

== Examples of topological spaces ==
{{seealso|List of topologies}}
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.

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.<ref>{{cite journal
 | last1 = Anderson | first1 = B. A.
 | last2 = Stewart | first2 = D. G.
 | doi = 10.2307/2037491
 | journal = Proceedings of the American Mathematical Society
 | jstor = 2037491
 | mr = 244927
 | pages = 77–81
 | title = <math>T_1</math>-complements of <math>T_1</math> topologies
 | volume = 23
 | year = 1969}}</ref>

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>(\alpha, \beta),</math> <math>[0, \beta),</math> and <math>(\alpha, \gamma)</math> where <math>\alpha</math> and <math>\beta</math> are elements of <math>\gamma.</math>

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

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

=== Topology from other topologies ===
{{split portions|section=y|Vietoris topology|Fell topology|date=June 2024}}
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. This construction is a special case of an [[initial topology]].

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. This construction is a special case of a [[final topology]].

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.

=== 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)#Definitions|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.

=== Topology from 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 ring]]s, [[topological field]]s and [[topological vector space]]s over the latter. [[Local field]]s are topological fields important in [[number theory]].

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.

=== 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 preorder'' (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]].

=== Topology from other structure ===
If <math>\Gamma</math> is a [[Filter (set theory)|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.

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]].

[[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>