Can Topological Spaces Have Properties Similar to Manifolds?

[trees and plants]

Hello. Could we define a topological space that locally resembles a riemannian manifold or another manifold like a complex manifold, or a Hermitian manifold near each point? Could it have interesting properties and theorems? Thank you.

 

[fresh_42]

This does not make much sense. A manifold is locally flat. So a manifold which is locally a manifold is still locally flat. You cannot iterate this procedure in a meaningful way.

Since the question is answered and of not much meaning, this thread is closed.