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Theorem 25.3 in Munkres proved that a space is locally connected if the components of every open subset is open.

What obviously follows from the theorem is that the components of a locally connected space is open. However, is there a nontrivial counterexample (in this case, disconnected space) where the components are open but the space is not locally connected?

cicolus
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  • This question has already been asked and answered here - Hope this helps you! https://math.stackexchange.com/questions/2589358/any-example-of-a-connected-space-that-is-not-locally-connected – Sundar Apr 17 '22 at 01:05
  • @Dr.Sundar The question you linked asked a different question -- I asked about spaces with open components and the question you linked asked about connected spaces. But I can see that is a trivial example of my question so I updated my question to reflect my intention. Thanks! – cicolus Apr 17 '22 at 01:22
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    Just take a disjoint union of a connected example and something else. – Eric Wofsey Apr 17 '22 at 01:42

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Of course. Pick $X\sqcup \{*\}$ for any connected but not locally connected $X$.

freakish
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