TITLE: locally self-similar topologies
QUESTION [6 upvotes]: 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?

REPLY [5 votes]: The Cantor set is indeed an example: it’s homeomorphic to $\{0,1\}^\omega$, where $\{0,1\}$ has the discrete topology, and so is every basic open set in this product. Indeed, if $X$ is any discrete space, and $\kappa$ is any infinite cardinal, $X^\kappa$ has the property: every member of the obvious base for the product topology is clearly homeomorphic to $X^\kappa$. The Sierpiński carpet is a Cantor set.
The Cantor set is almost an example of a stronger property: it’s almost a space in which every non-empty open set is homeomorphic to the whole space. It actually has two kinds of non-empty open subset, compact ones, which are homeomorphic to the Cantor set, and non-compact ones, which are homeomorphic to the Cantor set minus a point and to the discrete union of $\omega$ copies of the Cantor set. 
The irrationals are an example of a space with the stronger property: by an old result of Alexandroff and Uryson they are the  unique topologically complete, separable, $0$-dimensional metric space that contains no non-empty compact open set, and all of these properties are inherited by non-empty open subsets. More generally, if $X$ is any infinite discrete space, $X^\omega$ is a metrizable space of weight $|X|$ with the stronger property.
If $\kappa$ is any infinite cardinal, $\{\kappa\setminus\alpha:\alpha<\kappa\}\cup\{\varnothing\}$ is a $T_0$ topology on $\kappa$ in which all non-empty open sets are homeomorphic. 
If $\lambda\le\operatorname{cf}\kappa$ is also an infinite cardinal, $\{U\subseteq\kappa:|\kappa\setminus U|<\lambda\}\cup\{\varnothing\}$ is a $T_1$ topology on $\kappa$ with the desired property; when $\lambda=\omega$ it’s simply the cofinite topology.