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While it is clear that a disjoint union of two $d$-manifolds is a $d$-manifold, it is not clear to me if the disjoint union of a $d_1$-manifold and a $d_2$-manifold is still a manifold and if yes under some conditions then what is its dimension?

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up vote 2 down vote accepted

Most definitions of manifolds exclude the disjoint union of manifolds of different dimension from being a manifold.

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In particular, to really answer your question you should tell us exacty what your definition of manidold is. – Mariano Suárez-Alvarez Aug 27 '12 at 21:20
it is the standard definition: every point has a neighborhood homeomorphic to $\mathbb R^d$ – palio Aug 27 '12 at 21:23
@palio: and what is $d$? (Is it allowed to vary or is it fixed?) – Qiaochu Yuan Aug 27 '12 at 21:24
$d$ is the dimension of the manifold – palio Aug 27 '12 at 21:25
Well, if you fix that definition, then no, a disjoint union of two manifolds of different dimension not a manifold (if $m\neq n$, then no neighborhood of a point of $E^n$ is homeomorphic to a neighborhood of $E^m$: this is an immediate consequence of the Theorem on Invariance of Domain) – Mariano Suárez-Alvarez Aug 27 '12 at 21:42

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