Homeomorphic characterization of the real line?












2














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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  • related post: mathoverflow.net/questions/76134/…
    – Josiah Park
    3 hours ago
















2














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 $mathbb{R}$?






gn.general-topology gt.geometric-topology







share|cite|improve this question
























  • related post: mathoverflow.net/questions/76134/…
    – Josiah Park
    3 hours ago














2












2








2







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 $mathbb{R}$?






gn.general-topology gt.geometric-topology







share|cite|improve this question















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 $mathbb{R}$?






gn.general-topology gt.geometric-topology




gn.general-topology gt.geometric-topology






share|cite|improve this question















share|cite|improve this question













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edited 3 hours ago









YCor

27.1k380132




27.1k380132










asked 3 hours ago









James Baxter

16811




16811












  • related post: mathoverflow.net/questions/76134/…
    – Josiah Park
    3 hours ago


















  • related post: mathoverflow.net/questions/76134/…
    – Josiah Park
    3 hours ago
















related post: mathoverflow.net/questions/76134/…
– Josiah Park
3 hours ago




related post: mathoverflow.net/questions/76134/…
– Josiah Park
3 hours ago










1 Answer
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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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    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.






    share|cite|improve this answer


























      5














      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.






      share|cite|improve this answer
























        5












        5








        5






        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.






        share|cite|improve this answer












        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.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 3 hours ago









        user131781

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