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This question already has an answer here:

Let $A$ be a path-connected subset of $\mathbb R^2$ such that the removal of any singleton from $A$ splits $A$ into two open connected components, each of which is path-connected.

Is $A$ necessarily homeomorphic to $\mathbb{R}$?

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marked as duplicate by Lee Mosher, Chris Godsil, Ben McKay, Community Jan 8 at 16:59

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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Ward has given the following characterization of the real line: a connected, locally connected separable metric space in which each point is a cut point, i.e., its removal splits the space into two connected subsets (Proc. London Math. Soc. 1936). This implies a positive answer to your question, assuming the set has more than one point.

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  • $\begingroup$ In order to apply this result, we need to establish the local connectednes sof $A$. And how to prove this fact? $\endgroup$ – Taras Banakh Jan 5 at 21:35

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