Topological property

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. 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 Cardinal function#Cardinal functions in topology 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 axiom 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.