Exercise 21, § 4

(a) Let $X$ be a completely regular space; now prove that, for $X$ to be locally connected, it is necessary and sufficient that for every normal open covering $\mathscr{R}$ of $X$, there exists a normal open covering $\mathscr{R}'$ finer than $\mathscr{R}$ and formed of connected open sets. (To see that the condition is sufficient, use Exercise 20 d). To see that it is necessary, consider a normal covering $\mathscr{R}$ of $X$, a sequence $(\mathscr{R}_n)$ of open coverings of $X$ such that $\mathscr{R}_1 = \mathscr{R}$, $V_{\mathscr{R}_{n+1}}^2 \subset V_{\mathscr{R}_n}$, and for every $n$, consider the covering $\mathscr{R}'_n$ formed by the connected components of the sets of $\mathscr{R}_n$.

(b) Let Y be a topological space, X a dense subspace of Y, y a point of Y, $(V_i)_{1 \leq i \leq n}$ a finite covering of X formed of connected sets. Prove that the union of the $\overline{V}_i$ which contain y is a connected neighbourhood of y.

(c) Let Y be a completely regular space, X a dense subspace of Y. Suppose that X is Weierstrassian (IX, p. 88, Exercise 20) and locally connected. Prove that Y is locally connected. (Let $y \in Y$, U an open neighbourhood of y in Y; there is a closed neighbourhood $V \subset U$ of y in Y such that $(U \cap X, (Y - V) \cap X)$ is a normal covering of X; use Exercise 21 a), Exercise 20 e) and Exercise 21 b)).