A connected space need not be locally connected; counterexamples include the comb space and broom space. Conversely, a locally connected space need not be connected; an easy counterexample is the union of two disjoint open intervals of the real line.
See alsoConnected im Kleinen, Locally Pathwise-Connected
This entry contributed by Margherita Barile
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Barile, Margherita. "Locally Connected." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/LocallyConnected.html