Mathematics Asked on February 8, 2021
I have encountered the following definition of a manifold:
A manifold $M$ is a space satisfying the following properties.
1- There exists a family of open neighborhoods $U_i$ together with continuous one-to-one mappings $f_i:U_irightarrow {mathbb R}^n$ with a continuous inverse for a number n.
2- The family of open neighborhoods covers the whole of $M$; i.e.
$bigcup_iU_i=M$
But doesn’t that mean the maps are bijections since they have inverses? If so, why didn’t write that instead of one-to-one?
Thanks for help in advance.
Item 1 is badly worded in a couple of ways, one of which is what you noticed. What I particularly notice is that the item 1 does not say whether $f_i$ needs to be surjective.
If the intention is that $f_i$ need not be surjective, then to define the relevant inverse map one should first specify the image $V_i = f_i(U_i) subset mathbb R^n$, and then say that $f_i^{-1} : V_i to U_i$ is continuous. But if that was indeed the intention then there's something missing: one must also specify that $V_i$ is an open subset of $mathbb R^n$.
If the intention is that $f_i$ is indeed surjective, then it would be nice if it said so.
I'll say that either of the above two alternatives is acceptable: they yield two equivalent definitions of a manifold, because of the fact that every open ball in $mathbb R^n$ is homeomorphic to $mathbb R^n$.
As for why the definition is poorly written in this fashion, I cannot help you. I would suggest asking the author/instructor/person-who-wrote-it, if possible.
Answered by Lee Mosher on February 8, 2021
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