- #1
Frillth
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Homework Statement
Suppose X ⊂ R^n is a k-dimensional manifold and Y ⊂ R^p is an l-dimensional manifold. Prove that:
X × Y = {[x,y] ∈ R^n × R^p : x ∈ X and y ∈ Y}
is a (k+l)-dimensional manifold in R^(n+p). (Hint: Recall that X is locally a graph over a k-dimensional coordinate plane in R^n and Y is locally a graph over an l-dimensional coordinate plane in R^p.)
Homework Equations
None known.
The Attempt at a Solution
Our book doesn't even give a really solid definition of a manifold, so I don't really know where to start.