topological space

== English == === Noun === topological space (plural topological spaces) (topology, formally) An ordered pair (X, τ), where X is a set and τ, called the topology, is a collection of subsets of X which satisfies certain axioms and whose elements are called the open sets (or alternatively, for a different set of axioms, the closed sets);(loosely) the set X. 2011, Jonathan A. Barmak, Algebraic Topology of Finite Topological Spaces and Applications, Springer, Lecture Notes in Mathematics 2032, page xi, Most of the spaces studied in Algebraic Topology, such as CW-complexes or manifolds, are Hausdorff. In contrast, finite topological spaces are rarely Hausdorff. A topological space with finitely many points, each of which is closed, must be discrete. ==== Usage notes ==== The set X {\displaystyle X} may be called the underlying set. The collection τ {\displaystyle \tau } may be called the topology. The topological space ( X , τ ) {\displaystyle (X,\tau )} may be described as the underlying set X {\displaystyle X} endowed with the topology τ {\displaystyle \tau } . ==== Hypernyms ==== mathematical space, space ==== Hyponyms ==== Cauchy space uniform space metric space normal space pointed space regular space sober space Hausdorff space suspension ==== Meronyms ==== topology open set underlying set ==== Translations ==== === Further reading === Topology on Wikipedia.Wikipedia Specialization (pre)order on Wikipedia.Wikipedia Spectral space on Wikipedia.Wikipedia Topological space on Encyclopedia of Mathematics