I just have started to learn basic topology and came across a question in my class note: Suppose that $U$ is a compact Hausdorff topological space and let $M$ be a countable infinite subset of $U$. Let $f:U\to V$ be a continuous function, where $V$ is a metric space. We know that $M$ must have a cluster point say $m$ in $U$.
What I am confused about is that :
There must exists a sequence $(u_n)$ in $M$ such that $f(u)$ is a sub-sequential limit of $(f(u_n))$.
What happens if we replace the compact by locally compact? What is the function of $U$ being Hausdorff?
Thanks in advance.