Definition 4

A topological space $X$ is said to be locally connected if each point of $X$ has a fundamental system of connected neighbourhoods.

\* We shall see in Chapter IV, § 2, no. 5, that the real line is a locally connected space. \* The existence, at each point $x$ of a space $X$, of one connected neighbourhood of $x$ by no means implies that $X$ is locally connected. In particular, $X$ can be connected but not locally connected (Exercises 2 and 13). Conversely, a space can be locally connected but not connected (e.g. a discrete space which contains more than one point).