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I am looking some reference, if it exist, that generalized the Moore`s characterization of the 2-sphere. To be more precise, Moore characterized 2-sphere by these two axioms: A space X is a 2-sphere if

  1. Jordan domain axiom: For every simple closed curve C, the complement X-C consists of two connected components called Jordan domains bounded by C. Moreover, the boundary of Jordan domain bounded by C is exactly C.
  2. Basic axiom There is a countable basis of topology in X consisting of Jordan domains.

I took this two axioms from the paper Moore`s Theorem - V. Timorin. In this same paper, the author says that R.H. Bing gave a strong version of this characterization: a compact, connected, locally connected Hausdorff space X with more than one point is homeomorphic to the 2-sphere, provide that no embedded $\mathbb{S}^0$ separates X, and all embedded $\mathbb{S}^1$ separate X.

My question is: Is there any such characterization for closed surfaces of genus g?

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  • $\begingroup$ If you may apply Bing's characterization locally then you get a characterization of a surface. After this you may add something that distinguishes a surface of genus $g$. Say any $g$ disjoint circles separate while there are $g-1$ disjoint circles that do not separate. $\endgroup$ – Anton Petrunin Dec 7 '18 at 3:41

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