Prove or disprove : limit point compact hausdorff space imply compact space?

Question

Prove or disprove : limit point compact hausdorff space imply compact space?.

I think that it is not true. Because we know that the every product of compact space is compact and we know that product of two limit point compact space and hausdorff is not limit point compact. (note that the product of two hausdorff space is hausdorff space.)

Answer 1

$\omega$ (the first uncountable ordinal) is limit point compact and Hausdorff with the order topology, but not compact.

Answer 2

The product fact is indeed enough: if we have $X$ and $Y$ that are limit point compact Hausdorff and such that $X \times Y$ is not limit point compact, we know that neither $X$ nor $Y$ is compact (as the product of a limit point compact space and a limit point compact $k$-space (which includes all compact Hausdorff spaces) is limit point compact. But this is quite non-trivial (the easiest construction I know involves subspaces of the Cech-Stone compactification constructed by transfinite recursion).

There are easier examples: $\omega_1$, the first uncountable ordinal (which Munkres calls $\Omega$) is sequentially compact, limit point compact, locally compact, hereditarily normal (so much more than Hausdorff) but not compact.

Another example is $\Sigma_{\omega_1} [0,1] = \{ f \in [0,1]^{\omega_1}: |f^{-1}[(0,1]]| \le \aleph_0 \}$, the so-called Sigma-product of uncountably many copies of $[0,1]$.