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,
