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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?

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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).

With the weaker notion of "poset stratification" as in Lurie Appendix A.5, you can also do some finagling with the power set of a countably infinite poset (the strata can still be topological spaces, but their dimensions may not correspond to their stratum order).

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    $\begingroup$ Do you think the same can be said for a compact space? $\endgroup$ – gian Nov 10 '17 at 4:36
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    $\begingroup$ @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. $\endgroup$ – Jānis Lazovskis Nov 10 '17 at 4:46

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