in topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. that is, a property of spaces is a topological property if whenever a space x possesses that property every space homeomorphic to x possesses that property. informally, a topological property is a property of the space that can be expressed using open sets. a common problem in topology is to decide whether two topological spaces are homeomorphic or not. to prove that two spaces are not homeomorphic, it is sufficient to find a topological property which is not shared by them. ==common topological properties== === cardinal functions === * the cardinality |x| of the space x. * the cardinality &tau;(x) of the topology of the space x. * weight w(x), the least cardinality of a basis of the topology of the space x. * density d(x), the least cardinality of a subset of x whose closure is x. === separation === for a detailed treatment, see separation axiom. some of these terms are defined differently in older mathematical literature; see history of the separation axioms. * t0 or kolmogorov. a space is kolmogorov if for every pair of distinct points x and y in the space, there is at least either an open set containing x but not y, or an open set containing y but not x. * t1 or frÃ©chet. a space is frÃ©chet if for every pair of distinct points x and y in the space, there is an open set containing x but not y. (compare with t0; here, we are allowed to specify which point will be contained in the open set.) equivalently, a space is t1 if all its singletons are closed. t1 spaces are always t0. * sober. a space is sober if every irreducible closed set c has a unique generic point p. in other words, if c is not the (possibly nondisjoint) union of two smaller closed subsets, then there is a p such that the closure of {p} equals c, and p is the only point with this property. * t2 or hausdorff. a space is hausdorff if every two distinct points have disjoint neighbourhoods. t2 spaces are always t1. * t2Â½ or urysohn. a space is urysohn if every two distinct points have disjoint closed neighbourhoods. t2Â½ spaces are always t2. * completely t2 or completely hausdorff. a space is completely t2 if every two distinct points are separated by a function. every completely hausdorff space is urysohn. * regular. a space is regular if whenever c is a closed set and p is a point not in c, then c and p have disjoint neighbourhoods. * t3 or 
Homeomorphisms Properties_of_topological_spaces