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. ∎

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