Hausdorffness is refining-preserved

From Topospaces
Jump to: navigation, search
This article gives the statement, and possibly proof, of a topological space property (i.e., Hausdorff space) satisfying a topological space metaproperty (i.e., refining-preserved property of topological spaces)
View all topological space metaproperty satisfactions | View all topological space metaproperty dissatisfactions
Get more facts about Hausdorff space |Get facts that use property satisfaction of Hausdorff space | Get facts that use property satisfaction of Hausdorff space|Get more facts about refining-preserved property of topological spaces

Statement

Property-theoretic statement

The property of topological spaces of being a Hausdorff space

Statement with symbols

Suppose (X,\tau) is a topological space and \tau' is a finer topology on X than \tau.

Proof

Given: A Hausdorff topological space (X,\tau), and a topology \tau' on X that is finer than \tau

To prove: (X,\tau') is Hausdorff

Proof: We need to show that for points x \ne y in X, there exist open sets U,V in the topology \tau' such that x \in U, y \in V, and U \cap V is empty.

Since \tau gives a Hausdorff topology, we can find open sets U,V in the topology \tau, such that x \in U, y \in V and U \cap V is empty. And since \tau' is finer than \tau, the sets U,V satisfy the condition in \tau' as well.