Thread: Pre-image open or not? View Single Post
P: 1,172
 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

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-1os-1.

Like Zhentil said, if there is y0 with f(x,y0)>a , then , by

(assumed) continuity of f, there is a ball B(x,y0) 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).