I have a question about an exercise in Munkres, $\displaystyle \S 28, 3.$

At first I thought I had proven some of these things, but I've seen some counterexamples online and I'm not familiar with some of the spaces used.

Let X be limit point compact.
(b) If A is a closed subset of X, does it follow that A is limit point compact?
(c) If X is a subspace of the Hausdorff space Z, does it follow that X is closed?

For (b), someone gave the example of the closed subset $\displaystyle [0,1]$ in the lower limit topology. This subset is not limit point compact (we consider the subset $\displaystyle (1-\frac{1}{k})_{k=1}^\infty$), but the original space isn't either. Am I missing something?

For (c), someone suggested that $\displaystyle [0,\omega_1)$ is a not closed, limit point compact subset of the Hausdorff space $\displaystyle [0,\omega_1]$, where $\displaystyle \omega_1$ is the first uncountable limit ordinal. I haven't worked with these ordinals much before, and I don't have much intuition on it. Could anyone be of help?