[: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 $X$ Referenced by: