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This question is in the spirit of this question Does every non-singleton connected metric space $X$ contains a connected subset (with more than one point) which is not homeomorphic with $X$? ; Fixing the convention that whenever we will talk about connected spaces or subsets we will talk about such with more than one point ; it is easy to observe that $\mathbb Z$ with co-finite topology fails to have the property mentioned in the question i.e. the space is connected and every proper connected subset is homeomorphic to the whole space . Now this counter example space is $T_1$ but not Hausdorff , nor regular , nor normal . So it is natural to ask : Does there exist a connected Hausdorff ( resp. for Regular , $T_3$ , Normal , $T_4$ ) space such that every proper connected subspace of it is homeomorphic to the whole space ?

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    It's not a counter example, but an example: the integers in the cofinite topology has the interesting property that it is connected and every non-empty connected subset of it is homeomorphic to the whole space. Now you wonder whether examples with higher separation properties exist as well. The integers in the cofinite topology do have the property in the title. – Henno Brandsma Jan 09 '16 at 16:36
  • @HennoBrandsma : They do have the property in the title of "this question " but not of the "other question " concerning metric spaces ; but anyways it's ok that you got the spirit of the question . Have you any other examples with higher separation properties ? –  Jan 10 '16 at 04:36
  • If you change the condition from "proper connected subspace" to "proper connected open subspace", then $\mathbb{R}$ is an example (it's Hausdorff, regular, T3, normal and T4). https://math.stackexchange.com/questions/239063/intervals-are-connected-and-the-only-connected-sets-in-mathbbr – hasManyStupidQuestions Mar 06 '22 at 13:06

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