subsets of that have open inverse images under In other words, the quotient topology is the finest topology on for which is continuous. A common example of a quotient topology is when an equivalence relation is defined on the topological space The map is then the natural projection onto the set of equivalence classes. The Vietoris topology on the set of all non-empty subsets of a topological space named for Leopold Vietoris, is generated by the following basis: for every -tuple of open sets in we construct a basis set consisting of all subsets of the union of the that have non-empty intersections with each The Fell topology on the set of all non-empty closed subsets of a locally compact Polish space is a variant of the Vietoris topology, and is named after mathematician James Fell. It is generated by the following basis: for every -tuple of open sets in and for every compact set the set of all subsets of that are disjoint from and have nonempty intersections with each is a member of the basis. Classification of topological spaces 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, compactness, and various separation axioms. For algebraic invariants see algebraic topology. Topological spaces with algebraic structure For any 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 groups, topological vector spaces, topological rings and local fields. Topological spaces with order structure Spectral: A space is spectral if and only if it is the prime spectrum of a ring (Hochster theorem). Specialization preorder: In a space the specialization preorder (or canonical preorder) is defined by if and only if where denotes an operator satisfying the Kuratowski closure axioms. See also Complete Heyting algebra – The system of all open sets of a given topological space ordered by inclusion is a complete Heyting algebra. Citations Bibliography Bredon, Glen E., Topology and Geometry (Graduate Texts in Mathematics),