Question about Frechet compactness

[facenian]

Homework Statement

Let $X$ be a limit point compact space. If $X$ is a subspace of the Housdorff space $Z$, does it follow that $X$ is closed in Z?

Homework Equations

A space $X$ is said to be limit point compact if every infinite subset of $X$ has a limit point.

The Attempt at a Solution

If $X$ is finite it is closed so suppose it is infinite then it has a limit point in $X$ however this does not exclude the posibility of $X$ having a limit point in $Z$ outside $X$ which implies that it is not closed in $Z$.
Is this correct? In which case a counter example is needed.

 

[facenian]

The statement is false. Take $Z = \mathbb{R}$ with the usual topology. Then $Z$ is Hausdorff as it is metrizable and $X = (0,1)$ is not closed in $Z$, while it is limit point compact as subspace (by the Bolzano-Weierstrass theorem)

The problem is that this is not a counter example. The infinite set $\{1/(n+1):n\in Z^+\}\subset (0,1)$ does not have a limit point in (0,1) so this space is not limit point compact. Given the Heine-Borel theorem for $R^n$ a counter example can exists only outside $R^n$ with the usual topology or the fact that limit point compactness and compactnes are equivalent in a metric space.

 

[member 587159]

The problem is that this is not a counter example. The infinite set $\{1/(n+1):n\in Z^+\}\subset (0,1)$ does not have a limit point in (0,1) so this space is not limit point compact. Given the Heine-Borel theorem for $R^n$ a counter example can exists only outside $R^n$ with the usual topology or the fact that limit point compactness and compactnes are equivalent in a metric space.

You are right. This has been a long day for me haha. Currently studying for a functional analysis exam. I deleted my post so that it seems that you didn't get any responses (I will delete this post too)

 

[facenian]

You are right. This has been a long day for me haha. Currently studying for a functional analysis exam. I deleted my post so that it seems that you didn't get any responses (I will delete this post too)

I believe that wrong answers are also instructive though and it is not good to delete them since they may help elucidating a correct answer.

 

[member 587159]

I believe that wrong answers are also instructive though and it is not good to delete them since they may help elucidating a correct answer.

I already deleted it, but it is still visible in your post. So it's up to you.

Anyways, this might be of interest to you:

https://math.stackexchange.com/ques...-of-a-hausdorff-space-necessarily-closed?rq=1

 

[facenian]

I already deleted it, but it is still visible in your post. So it's up to you.

Anyways, this might be of interest to you:

https://math.stackexchange.com/ques...-of-a-hausdorff-space-necessarily-closed?rq=1

Thank you. In this link it is claimed a standard counter example is given.