Exercise 22, § 4

Let $X$ be a regular space; now prove that the following three properties are equivalent: α) $X$ is locally connected and semi-compact (IX, p. 93, Exercise 13). β) For every finite open covering of $X$, there exists a finer finite open covering, formed of connected sets. γ) For every finite open covering of $X$, there exists a finer finite covering (not necessarily open) formed of connected sets. (To see that γ) implies that $X$ is locally connected, reason as in Exercise 21 c). To see that α) implies β), argue by contradiction: if $(U_j)$ is a finite open covering of $X$, form by induction on j, an open covering $(V_\lambda)$ each element of which is a connected component of one of the $U_j$, and where none of the $V_\lambda$ is contained in the union of the others; if $(V_\lambda)$ is infinite, deduce that $X$ would not be semi-compact).