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Feb 14 at 18:14 comment added Yasha I see, in our case the relevant space $Diff_c(M)$ is separable. So you have proved it has the homotopy type of a countable CW complex. I don't know if you want to add this to your answer.
Feb 14 at 2:26 comment added Tyrone @Yasha I'm glad it was useful to you. Separable here just means having a countable dense set.
Feb 13 at 23:02 vote accept Yasha
Feb 13 at 22:39 comment added Yasha Thank you for this detailed answer. What is a separable stratifiable space in this context?
Feb 12 at 7:05 history answered Tyrone CC BY-SA 4.0