Section 28: Problem 3 Solution
Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
(a) No. If
is compact the image is compact, so if we believe the statement is wrong, we need to look for
which is limit point compact but not compact. Consider
in the Example 1. The projection on the first coordinate is continuous (it can also be thought of as the quotient space obtained by identifying all points in
) but maps the limit point compact space to the not limit point compact set
.(b) Yes. An infinite subset of
has a limit point in
which is a limit point of
as well, i.e. it is in
.(c) No. Once again for the counterexample we need a limit point compact space which is not compact. Now the Example 2 works better:
. Note that
not only Hausdorff but also compact (and, hence, limit point compact).