Call a topology "locally self-similar" if it has a basis in which each open set is homeomorphic to the entire space. What topologies have this property?
So far, I have the following list:
- Any set with the indiscrete topology (the whole space is the unique neighborhood of any point).
- The real numbers.
- The rational numbers (as a subspace of the real numbers).
- Probably the Cantor set or something similar (I'm not sure whether the endpoints look locally like the other points).
- Probably the Sierpinski carpet and lots of similar spaces.
- Probably the irrational numbers.
- Any finite product of spaces with this property.
Anything else? Is it possible to classify these spaces in any interesting way?