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 $y_{1}\neq y_{2}$ . 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
Canonical name	HausdorffPropertyIsHereditary
Date of creation	2013-03-22 15:22:27
Last modified on	2013-03-22 15:22:27
Owner	georgiosl (7242)
Last modified by	georgiosl (7242)
Numerical id	8
Author	georgiosl (7242)
Entry type	Theorem
Classification	msc 54D10