Quote by Jooolz
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,

Well, let's see. Let me review for next time I teach point set topology, so you can
combine it with Zhentil's answer:
What is g
^{1}(a,b)? it is the collection of all x such that there is a y in Y
with a<f(x,y)<b.
We have g=Sof(x,y) , where f:XxY→f(XxY), and S:f(XxY)→ℝ , and g
^{1}:=
f
^{1}os
^{1}.
Like Zhentil said, if there is y
_{0} with f(x,y
_{0})>a , then , by
(assumed) continuity of f, there is a ball B(x,y
_{0}) where f(x,y)>a . This
gives you openness in the subspace f(XxY). Now, compose with f
^{1}, to
get an open set in XxY, by assumed continuity of f(x,y).