Exercise 19, § 4

(a) Let $X$ be a completely regular space; now prove that a point $a$ of $\beta X - \upsilon X$ cannot admit a fundamental system of connected neighbourhoods in $\beta X$. Observe that there exists a function $f \in C(X; \mathbf{R})$ which tends to $+\infty$ at the point $a$. For $i = 0, 1, 2, 3$, let $Z_i$ be the set of the $x \in X$ such that $n \leq f(x) \leq n+1$ for an $n \equiv i \pmod{4}$. If, for example, $a$ is adherent to $Z_3$, observe that $Z_1$ and $Z_3$ are normally separated in $X$ (IX, p. 87, exerc. 15), and consequently that there exists an open neighbourhood $U$ of $a$ in $\beta X$ such that $Z_1 \cap U = \varnothing$. Now prove that it is not possible that there exists a connected neighbourhood $U' \subset U$ of $a$, by using IX, p. 87, exerc. 16.

(b) Deduce from a) that, in order that $\beta X$ be locally connected, it is necessary and sufficient that $X$ be locally connected and weierstrassian (cf. IX, p. 105, exerc. 21).