I was thinking manifold hypothesis which states that real-world high-dimensional data lie on low-dimensional manifolds embedded within the high-dimensional space.
But I guess we can't choose latent space dimension arbitrary. Like a data set $X$ lying on the 2-dimensional sphere $S^2$ embedded in the ambient space $R^n$ where $n \geq 3$. It is well known that there exist no homeomorphic maps between $S^2$ and an open domain on $R^2$.
So, I was wondering, is there any research done on this field where some necessary or sufficient condition was given for existence of homeomorphic map or conclude no homeomorphic map exist like above? It will be a great help if anyone cite those or redirect me some resources. I was interested in both theoretical and application (on deep learning) field.
I have some basic knowledge of differential geometry and tensor calculus.