# Uncountability of admissible topological stratifications

I'm not sure if this is trivial. Can someone please provide an example of a space which admits uncountably many topological stratifications? How about up to homeomorphism?

You can take the real line as $(-\infty,x) \cup (x,\infty)$ stratified higher than $\{x\}$, for every real number $x$. But these certainly are all homeomorphic.
Judging by this question and the Whitney embedding theorem, there are uncountably many different 2-dimensional strata of $\mathbf R^4$ (as there are uncountably many non-compact surfaces).
• @gian I am not sure, but I would guess yes, maybe by using the stereographic projection of $\mathbf R^4$ onto $S^4$. Then you probably wouldn't have surfaces as strata, but more general topological spaces. – Jānis Lazovskis Nov 10 '17 at 4:46