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Let $(X,d)$ be a metric space and $(Y,d_Y)$ a subspace with $Y \subset X$

Prove that:

A subset $F$ of $Y$ is closed in $Y$ if and only if $F=Y\cap H$, for some closed subset $H$ of $X$

I'm having trouble proving the first implication (so assuming that $F$ is a closed subset in $Y$ and proving the latter).

The user egreg seems to answer this here (the second answer): Show that $F\subset Y$ is closed in $Y$ iff $F=Y\;\cap\;H$ where $H\subset X$ is closed in $X$.

But I'm not convinced by this statement:

Let $x\in F$; then $x\notin Y\setminus F$, so $x\notin Y\cap A$ and therefore $x\notin A$. So $x\in H$.

In the case that $A \subset Y$ we can simply conclude that $x \notin A$, otherwise it wouldn't necessarily hold.

How can I prove this?

Community
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Lorkus
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1 Answers1

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This is known as the inheritance principle in point-set topology. That is, a subspace of a metric space $X$ inherits its closeds from the parent space.

Suppose that $F\subset Y$ is closed in $Y$. Consider $\operatorname{cl}_X{F}\subset X$, the closure of $F$ in $X$. Then $\operatorname{cl}_X{F}$ is a closed subset of $X$ and consists of $F$ together with its limit points in $X$. None of $F$'s limit points lies in $Y\setminus F$ since $F$ is closed in $Y$. Hence, $Y\cap\operatorname{cl}_X{F} = F$ as desired.

Alex Ortiz
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