Section 18: Problem 2 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
Suppose that
is continuous. If
is a limit point of the subset
of
, is it necessarily true that
is a limit point of
?
No, for example,
can be a constant function.
To elaborate a bit. Suppose we wanted to prove this statement true. Having an open neighborhood
of
we would want to show that it intersects
in a point different from
.
is an open neighborhood of
, hence, there is some
, and
. By this, we only showed that
, but it can be the case that for every
,
. So, if
has a neighborhood
such that
, then
is not a limit point of
.