Do continuous functions preserve limit point compactness when the spaces are Hausdorff?

Question

This is based on the problem in Munkres.

Let $X$ be a limit point compact Hausdorff space and $Y$ a Hausdorff space. Let $f: X \to Y$ be continuous. Is $f(X)$ limit point compact in $Y$?

Answer 1

For Hausdorff spaces, limit point compact is just equivalent to countably compact (every countable open cover has a finite subcover). Then the same proof as for compactness will show that it is indeed true (pull back a countable open cover of $f[X]$, etc).