Can Topological Spaces Have Properties Similar to Manifolds?

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SUMMARY

The discussion centers on the possibility of defining a topological space that locally resembles Riemannian, complex, or Hermitian manifolds. It concludes that while the idea is intriguing, a manifold's local flatness limits the meaningful iteration of this concept. The thread was ultimately deemed to lack substantive value, leading to its closure.

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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.
 
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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.
 

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