1. ## Continuous function

Suppose fi is continuous from a topological space X to the real numbers,how to prove that g=sup{fi:i=1...n} is also continuous?

2. Originally Posted by weidingchang
Suppose fi is continuous from a topological space X to the real numbers,how to prove that g=sup{fi:i=1...n} is also continuous?
It is quite clear to show that a function is continuous we must only show that it's continuous with respect to some open subbase for the codomain. So, in particular since if $K=\left\{(a,\infty):a\in\mathbb{R}\right\}$ and $L=\left\{(-\infty,b):b\in\mathbb{R}\right\}$ then $K\cup L$ forms an open subbase for $\mathbb{R}$ we must only check the cases for elements of this set.

If $E\in K$ then $E=\left(a,\infty\right)$ and so $g^{-1}\left(E\right)=\left\{x\in X:\max_{1\leqslant j\leqslant n}f_j(x)>a\right\}=\bigcup_{j=1}^{n}\left\{x\in X:f_j(x)>a\right\}$ and since this is a union of open sets (since each $f_j$ is continuous) it follows that $g^{-1}\left(E\right)$ is continuous. The case for $E\in L$ is analogous.

3. ## Another question

How to prove that fg is continuous if both f and g are continuous from a topological space X to the real numbers?Thanks a lot...

4. Originally Posted by weidingchang
How to prove that fg is continuous if both f and g are continuous from a topological space X to the real numbers?Thanks a lot...
Hint hint:

Spoiler:

Try it yourself and let us know what you are having trouble with.

5. Originally Posted by weidingchang
How to prove that fg is continuous if both f and g are continuous from a topological space X to the real numbers?Thanks a lot...

Try writing down the relation between the inverse images of f,g and fg.