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when we say that :"topological space M admits (for example)30 differential structure." is it means that for some fixed topology, M admits 30 differential structure or for different topology?

mja
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  • When you say "topological space $M$..." this means that you have fixed a topological structure $T$ on a certain set $X$: $M=(X,T)$. Notice that you did not say "a set $M$ admits ..." (the latter would allow for different topologies on $M$). – Moishe Kohan Jun 24 '17 at 11:36

1 Answers1

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This simply means that the $30$ structures are homeomorphic, but not diffeomorphic $-$ i.e., they have the same topology, but they have different smooth functions.

Yousuf Soliman
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