Hausdorffness is refining-preserved
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 with symbols
Suppose is a topological space and is a finer topology on than .
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.