Proposition 11

A necessary and sufficient condition for a space $X$ to be locally connected is that every component of an open set in $X$ is open in $X$.

The condition is sufficient, since the component of $x$ relative to an open neighbourhood of $x$ is then a neighbourhood of $x$ in $X$.

Conversely, let $A$ be an open subset of a locally connected space $X$, let $B$ be a component of $A$, and let $x \in B$. Let $V$ be a connected neighbourhood of $x$ contained in $A$; by the definition of components, $V$ is contained in $B$; hence $B$ is open in $X$ (§ 1, no. 2, Proposition 1).

The components of a locally connected space $X$ therefore form a partition of $X$ into open sets, and hence $X$ is the sum (§ 2, no. 4, Example 3) of its components.