# Hausdorffness is refining-preserved

From Topospaces

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)

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## Statement

### Property-theoretic statement

The property of topological spaces of being a Hausdorff space

### Statement with symbols

Suppose is a topological space and is a finer topology on than .

## Proof

**Given**: A Hausdorff topological space , and a topology on that is finer than

**To prove**: is Hausdorff

**Proof**: We need to show that for points in , there exist open sets in the topology such that , and is empty.

Since gives a Hausdorff topology, we can find open sets in the topology , such that and is empty. And since is finer than , the sets satisfy the condition in as well.