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Jan31-12, 09:02 PM
P: 11
Hi all,

I am struggling with the following:

If X and Y are topological spaces. and f: X x Y → ℝ is a continuous function (product topology on X x Y, Euclidean topology on ℝ)

Let g: X → ℝ defined by g(x) = sup { f(x,y) | y in Y }

Then: If A=(r, ∞) for r in ℝ, g-1(A) is open. And If A=(-∞, t) for t in ℝ, g-1(A) is not always open.

Why is that? How can I know if g-1(A) is open or not if I dont know anything about X??

Does anyone have an idea?

kind regards,
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