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Homework Statement
The lemma sets out to show that if A in R^n is compact and x_0 is in R^m, then A x {x_0} is compact in R^n x R^m.
They say, "Let [itex]\mathcal{U}[/itex] be an open cover of A x {x_0} and
[tex]\mathcal{V}=\{V\subset \mathbb{R}^n:V=\{y:(y,x_0)\in U\}, \ \mbox{for some} \ U\in \mathcal{U}\}[/tex]
Then [itex]\mathcal{V}[/itex] is an open cover of A."
How do they know that given some U in [itex]\mathcal{U}[/itex], the associated V is open?
Edit: In fact, consider the following counter example: Let n=m=1, A=[-½,½], x_0=0. Then A x {x_0} is just the segment [-½,½] considered in the R² plane. Let [itex]\mathcal{U}=\{B_n(0,0)\}_{n\in\mathbb{N}}[/itex] (the collection of open balls centered on the origin of radius n). Then, let V_n be the set V associated with B_n(0,0) as described above, i.e. [itex]V_n=\{y\in\mathbb{R}:(y,0)\in B_n(0,0)\}[/itex]=[-½,½], a set that is not open in R. ah!
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