lower semi continuous function

• Mar 19th 2011, 05:45 AM
asterias
lower semi continuous function
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!
• Mar 19th 2011, 11:04 AM
FernandoRevilla
The result is also valid for the supremum $f=\sup \{f_i:\;i\in I\}$ of lower semicontinuous functions $f_i:X\to \mathhbb{R}$ , $(X,T)$ topological space .

Hint:

$G_i=f_i^{-1}(\alpha,+\infty)\in T$ for all $i\in I$ so, $G=\bigcup\limit_{i\in I}G_i \in T$

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• Apr 3rd 2011, 03:09 AM
asterias
Thank you very much for your help :-)