Let $A,B \subseteq \mathbb R^k$ be disjoint smooth manifolds of different dimensions , then is it rue that $A \cup B $ cannot be a smooth manifold ?
Is it true that , in a Euclidean space , union of two disjoint smooth manifolds of different dimensions can never be a manifold?
2
$\begingroup$
manifolds
smooth-manifolds
-
1What is your definition of (topological or smooth) manifold. Actually, some authorsdo allow "mixed" dimensions. – 2017-02-12
1 Answers
4
In ${\mathbb R}^2$ let
$$A:=\bigl\{(x,y)\,\bigm|\,1
-
1+1 Didn't think of that. Even if one requires that manifolds be pure and connected, $A=$open disk minus one point and $B=$that point would make a similar great example – 2017-02-12
-
0@HagenvonEitzen : can a single point be a smooth manifold ? – 2017-02-12
-
1@SaunDev Okay, everything $\times \Bbb R$ if you prefer :) – 2017-02-12