I working through the proof of the stack of records theorem and am stuck on this fact.
The following is taking from http://www.math.sjsu.edu/~simic/Spring09/Math213/final.213.09.soln.pdf page 35.
$f^{−1}(y)$ is a regular submanifold of $M$ of dimension zero, i.e.,$f^{−1}(y)$ is a discrete subset of $M$. This implies that $f^{−1}(y)$ is at most countable. If $f^{−1}(y)$ is infinite, then by compactness of $M$, $f^{−1}(y)$ has an accumulation point, which must be in $f^{−1}(y)$, since $f^{−1}(y)$ is closed. But this contradicts the fact that $f^{−1}(y)$ is discrete.Therefore, $f^{−1}(y)$ is a finite set $\{x_1,...,x_N\}$.
Having an accumulation point means that every neighborhood of the accumulation point contains another point that is not the accumulation point. Why does this mean that the set is not discrete? Could it not be the case that there are only a discrete set of neighborhoods around the point?
A discrete space $X$ cannot have an accumulation point $x$ since the singleton $\{x\}$ is an open neigbourhood of $x$ which does not contain a point different from $x$.