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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. ∎

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

54D10*no label found*

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Apr 15

new question: Prove a formula is part of the Gentzen System by LadyAnne

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new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

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new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier

## 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 ✓