[:locally connected:] [:local connectedness:]
A topological space $X$ is called locally connected at $x$ if for each open $U \ni x$ there exists a connected neighborhood $V \ni x$ with $V \subseteq U$.
If $X$ is locally connected at every $x \in X$, we simply call it "locally connected".
Ex: splitting $\mathbb R$ into two disjoint intervals forms a locally connected (but not connected) space
Theorem (JRM:Top.2 §25.3). TFAE:
$X$ is locally connected
For each open $U \subseteq X$, every component of $U$ is open in $X$
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