A real-valued function $f:X \rightarrow \mathbb{R}$ is upper semicontinuous if for each $c \in \mathbb{R}$, its pre-image $f^{-1}(-\infty,c)$ is open in $X.$
In encyclopaedia, there is the following statement:
Let $\mathcal{F}$ be an arbitrary family of upper semicontinuous functions on a given topological space $X$. Then the function $F(x) = \inf_{f \in \mathcal{F}} f(x)$ is upper semicontinuous.
The article stated that the proof can be found in General Topology by Bourbaki, Chapter $5 – 10$. But I do not have access to that book.
In Wikipedia, similar statement appears.
Likewise, the pointwise infimum of an arbitrary collection of upper semicontinuous functions is upper semicontinuous.
I manage to prove the statement holds for countably many functions, its proof is similar to proving infimum of measurable functions is measurable, but I have no idea on how to prove for arbitrary case.
Best Answer
$$F(x)<c\iff\exists f\in\mathcal F [f(x)<c]$$ so that: $$F^{-1}(-\infty,c)=\bigcup_{f\in\mathcal F}f^{-1}(-\infty,c)$$ If the $f\in\mathcal F$ are upper semi-continuous then this is a union of open sets, hence is open.