Editing Topological space (section)

=== 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.