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?
Dimension of disjoint union of manifolds
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$\begingroup$
general-topology
manifolds
1 Answers
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Most definitions of manifolds exclude the disjoint union of manifolds of different dimension from being a manifold.
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0In particular, to *really* answer your question you should tell us exacty what your definition of manidold is. – 2012-08-27
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0it is the standard definition: every point has a neighborhood homeomorphic to $\mathbb R^d$ – 2012-08-27
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1@palio: and what is $d$? (Is it allowed to vary or is it fixed?) – 2012-08-27
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0$d$ is the dimension of the manifold – 2012-08-27
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0(or, more precisely, where the quantifier for $d$ put?...) – 2012-08-27
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0(Also: That is not the standard definition...) – 2012-08-27
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0well 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. – 2012-08-27
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1Well, 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) – 2012-08-27
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0are there different definitions of topological manifolds? – 2012-08-27