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

Property-theoretic statement

The property of topological spaces of being a Hausdorff space

Statement with symbols

Suppose $(X,\tau )$ is a topological space and ${\tau }^{\prime }$ is a finer topology on $X$ than $\tau$.

Proof

Given: A Hausdorff topological space $(X,\tau )$, and a topology ${\tau }^{\prime }$ on $X$ that is finer than $\tau$

To prove: $(X,{\tau }^{\prime })$ is Hausdorff

Proof: We need to show that for points $x\ne y$ in $X$, there exist open sets $U,V$ in the topology ${\tau }^{\prime }$ 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 ${\tau }^{\prime }$ is finer than $\tau$, the sets $U,V$ satisfy the condition in ${\tau }^{\prime }$ as well.