Hi,

I have a question about an exercise in Munkres,

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 in the lower limit topology. This subset is not limit point compact (we consider the subset ), but the original space isn't either. Am I missing something?

For (c), someone suggested that is a not closed, limit point compact subset of the Hausdorff space , where 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?

Regards,

Michael