Dimension of disjoint union of manifolds

Question

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?

score 4 · Accepted Answer · answered Aug 27 '12 at 21:14

4

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

answered Aug 27 '12 at 21:14

Mariano Suárez-Álvarez

135,076

In particular, to really answer your question you should tell us exacty what your definition of manidold is. – Mariano Suárez-Álvarez 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
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@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
(or, more precisely, where the quantifier for $d$ put?...) – Mariano Suárez-Álvarez Aug 27 '12 at 21:25
(Also: That is not the standard definition...) – Mariano Suárez-Álvarez Aug 27 '12 at 21:28
well i said it roughly but wikipedia says the same thing: A topological space $X$ is called locally Euclidean if there is a non-negative integer $n$ such that every point in $X$ has a neighborhood which is homeomorphic to the Euclidean space $E^n$ (or, equivalently, to some connected open subset of $E^n$).
A topological manifold is a locally Euclidean Hausdorff space.
– palio Aug 27 '12 at 21:39
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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-Álvarez Aug 27 '12 at 21:42
are there different definitions of topological manifolds? – palio Aug 27 '12 at 21:53
@palio and Mariano Suárez-Álvarez, This one, right? – Sep 11 '19 at 04:12

Dimension of disjoint union of manifolds

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