Hausdorff property is hereditary

Theorem 1.

A subspace of a Hausdorff space is Hausdorff.

Proof.

Let $X$ be a Hausdorff space , and let $Y$ be a subspace of $X$. Let $y_1,y_2\in Y$ where $y_1\ne y_2$.Since $X$ is Hausdorff, there are disjoint neighborhoods $U_1$ of $y_1$ and $U_2$ of $y_2$ $\in X$. Then$U_1\cap Y$ is a neighborhood of $y_1$ in $Y$ and $U_2\cap Y$ is a neighborhood of $y_2$ in $Y$, and $U_1\cap Y$ and $U_2\cap Y$ are disjoint.Therefore, $Y$ is Hausdorff.∎