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 $y_{1}\eq y_{2}$ .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.∎

单词	HausdorffPropertyIsHereditary
释义	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 $y_{1}\eq y_{2}$ .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.∎
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