Exercise 10, § 5

Let $X$ be a topological space which has a countable base $(U_n)$ and let $f$ be a real-valued function defined on $X$; $f$ is said to attain a strict relative maximum at a point $a \in X$ if there is a neighbourhood $V$ of $a$ such that $f(x) < f(a)$ for each $x \in V$ other than $x = a$. Show that the set $M$ of points of $X$, at which $f$ attains a strict relative maximum, is countable. (Consider the set of $U_n$ such that $f$ attains, at some point of $U_n$, a strict relative maximum equal to its least upper bound in $U_n$, and show that there is a mapping of this set onto $M$).