Let $X$ be a separable metric space. Suppose that $X$ contains no subspace homeomorphic to the plane. Is it necessarily true that the dimension of $X$ is $\leq 1$?
2026-03-25 09:27:05.1774430825
Does a space have dimension at most 1 if it contains no plane?
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No, there are totally disconnected separable metric spaces of any dimension $n$ (even $\infty$). And totally disconnected spaces do not contain any connected subspace except singletons...