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