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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?

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    There is also the right (or left) order topology on the real numbers, which is not only locally self-similar but self-similar (@BrianMScott's "stronger property"). The same construction seems to work for the rational and irrational numbers: make a new topology whose open sets are the existing open sets intersected with all $(x, \infty)$ for all real $x$. – 2012-09-21
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    Amazing question. I am sure there is some sufficient condition in terms of action of a semigroup (I can see how it arises in examples with reals, rationals, totally disconnected spaces, as well as some non-Hausdorff (albeit T₀) spaces not mentioned by original poster). Will think on it further. – 2014-11-09
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    Great, @IncnisMrsi, tell me what you find. – 2014-11-14

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