Hausdorffness is refining-preserved

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 $x\neq y$ 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.