Because $\mathbb{R}$ is homeomorphic to any open interval, and a subset of $\mathbb{R}$ is connected if and only if it is an interval, it has the unusual property of being homeomorphic to any of its proper open connected subsets. In particular, given any point in $\mathbb{R}$, all of its proper connected open neighborhoods are homeomorphic to $\mathbb{R}$ itself.
Additionally, any space with the trivial topology vacuously has this property, since it has no proper open subsets.
Question: Are there any other nontrivial spaces besides $\mathbb{R}$ with this unusual/idiosyncratic property? If so, has anyone ever studied this class of topological spaces and given them a name? If so, what is the name?
Note that $\mathbb{R}^n$ for $n \ge 2$ does not have this property (cf. this question or this one), $\mathbb{R}^2 \setminus \{0 \}$ is a counterexample (and one that shows we have to get algebraic topology involved for $n \ge 2$).
Best Answer
The space of integers with the cofinite topology is connected and has this property. Or the reals with the right ray topology.