Exercise 20, § 1

a) Let $X$ be a completely regular space, and let $a$ be a point of $X$. Then there exists a continuous function $f \geq 0$ on $X$ such that $f(a) = 0$ and $f(x) > 0$ whenever $x \neq a$, if and only if there is a sequence $(V_n)$ of neighbourhoods of $a$ in $X$ such that $\bigcap_n V_n = \{a\}$ [cf. Exercise 16 a)].

b) Let $X$ be a completely regular space, each point of which has a countable fundamental system of neighbourhoods. Show that, in the Stone-Čech compactification $\beta X$, the set $X$ is equal to the set of points which have a countable fundamental system of neighbourhoods in $\beta X$ [use a) and Exercise 12]. Let $X'$ be another completely regular space, each point of which has a countable fundamental system of neighbourhoods. Show that if the Stone-Čech compactifications $\beta X$ and $\beta X'$ are homeomorphic, then $X$ and $X'$ are homeomorphic.