If $Y$ is a Hausdorff space and $X$ which is a subspace of that space is limit point compact space . Then $X$ is closed.

Question

If $Y$ is a Hausdorff space and $X$ which is a subspace of that space is limit point compact space . Then $X$ is closed.

Can anyone give me a trivial counter example?

Answer 1

Let $Y$ be $\omega_1 + 1$ and $X$ is $\omega_1$, in the order topology. Here $\omega_1$ is the first uncountable ordinal, and $Y$ is its successor (one extra point).

Another example: let $Y$ be $[0,1]^\mathbb{R}$ in the product topology, and $X$ the limit point compact subset of all elements that are $0$ except for at most countably many coordinates. ( A $\Sigma$-product, this is called). This $X$ is dense and not closed in $Y$.