Hausdorffness is refining-preserved

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.