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# 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. ∎

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Reference

## Mathematics Subject Classification

54D10*no label found*

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## Corrections

title and other by igor ✓

contains own proof by rspuzio ✓

improve first sentence by Mathprof ✓

proof supplied by Wkbj79 ✓

contains own proof by rspuzio ✓

improve first sentence by Mathprof ✓

proof supplied by Wkbj79 ✓