Hello !
I want to prove that the supremum of continuous functions is a lower semi continuous function .
Could someone help me please ?I tried to find on the internet something to help me ,but i had no luck!
Hello !
I want to prove that the supremum of continuous functions is a lower semi continuous function .
Could someone help me please ?I tried to find on the internet something to help me ,but i had no luck!
The result is also valid for the supremum $\displaystyle f=\sup \{f_i:\;i\in I\}$ of lower semicontinuous functions $\displaystyle f_i:X\to \mathhbb{R}$ , $\displaystyle (X,T)$ topological space .
Hint:
$\displaystyle G_i=f_i^{-1}(\alpha,+\infty)\in T$ for all $\displaystyle i\in I$ so, $\displaystyle G=\bigcup\limit_{i\in I}G_i \in T$
.