Hausdorff property is hereditary

Proof.

Let $X$ be a Hausdorff space , and let $Y$ be a subspace of $X$. Let $y_{1},y_{2}\in Y$ where $y_{1}\neq y_{2}$. 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. ∎

Title Hausdorff property is hereditary HausdorffPropertyIsHereditary 2013-03-22 15:22:27 2013-03-22 15:22:27 georgiosl (7242) georgiosl (7242) 8 georgiosl (7242) Theorem msc 54D10