In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spaces which is closed 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. Properties of topological properties A property is: Hereditary, if for every topological space and subset the subspace has property Weakly hereditary, if for every topological space and closed subset the subspace has property Common topological properties Cardinal functions The cardinality |X| of the space X. The cardinality τ(X) of the topology (the set of open subsets) 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 Note that 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 Fréchet. A space is Fréchet 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