# Is every open set a union of (any number of) sets that are homeomorphic to the entire space?

For what spaces can we say that every open set is a union of (any number of) (not necessarily disjoint) sets that are homeomorphic to the entire space?

In $\mathbb{R}$ it's true (even with a countable number of open intervals, and with disjoint unions - but I don't care for these conditions in this question). I think that in $\mathbb{R}^n$ it's also possible with open balls. What about in a metric space, or even a topological space?

I think I need to exclude finite spaces because for them a subspace won't be homeomorphic to the whole space because of different cardinality.

• In any normed linear space open balls are homeomorphic to the whole space and any open set is a union of open balls. – Kavi Rama Murthy Mar 17 '18 at 12:19
• In any infinite cofinite space, a non-empty open set is homeomorphic to the whole space. – Henno Brandsma Mar 17 '18 at 12:50
• You need to exclude closed surfaces, like spheres or tori, because there the only subset homeomorphic to the whole space is the whole space. Similarly for higher-dimensional closed manifolds (and also lower-dimensional --- the circle needs to be excluded). – Andreas Blass Mar 17 '18 at 18:14
• @AndreasBlass oh, that's the counterexample I was missing .... thanks! – Dan Mar 18 '18 at 0:25
• No proper open subset of $[0,1]$ is homeomorphic to $[0,1]$. – Kavi Rama Murthy Mar 19 '18 at 23:32