A few limit point compactness questions

[radou]

Homework Statement

Let X be limit point compact (i.e. any infinite subset of X has a limit point).

(a) if f : X --> Y is continuous, does it follow that f(X) is 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 in Z?

The Attempt at a Solution

(a) well, if both X and Y are metrizable, this holds for sure, since limit point compactness of X implies compactness of X, which implies compactness of f(X), which then implies limit point compactness of f(X). otherwise, I don't see why this should hold.

(b) no, since for example [0, 1] is a closed subset of R in the lower limit topology, and it is not limit point compact

(c) well, again, if X is metrizable, X is compact, and hence closed in Z, since Z is Hausdorff. otherwise, I don't have an idea why it would hold.

any suggestions?

 

[micromass]

for a) take {0,1} with the indiscrete topology. Then
$$p_2:\{0,1\}\times\mathbb{Z}\rightarrow \mathbb{Z}$$
should do the job.

b) Correct. The example of above should do the job to.

c) Correct me if I'm wrong, but $$[0,\omega_1 [$$ is a non-closed countably compact (hence limit point compact) subset of the Hausdorff space $$[0,\omega_1]$$.

 

[radou]

for a) take {0,1} with the indiscrete topology. Then
$$p_2:\{0,1\}\times\mathbb{Z}\rightarrow \mathbb{Z}$$
should do the job.

b) Correct. The example of above should do the job to.

c) Correct me if I'm wrong, but $$[0,\omega_1 [$$ is a non-closed countably compact (hence limit point compact) subset of the Hausdorff space $$[0,\omega_1]$$.

micromass, thanks for the reply.

For c) does [0, ω1[ represent [0, ω1>? Which space are your referring to? i.e. what exactly does ω1 stand for, any real number?

 

[micromass]

$$\omega_1$$ is the first uncountable limit ordinal. I believe it is called $$\Omega$$ in Munkres (this exercise is from Munkres, isn't it).

The space $$[0,\omega_1[$$ is the space $$S_{\Omega}$$ of Munkres (although I'm not hundred percent sure of this).