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.