An interesting question came up in our analysis class today.
Let $S= \{f:\Bbb R \rightarrow \Bbb R \cup \{\infty\} \ | \ \{f>c\} \text{ is open for each } c \in \Bbb R\}.$ If $A$ is a nonempty indexing set and for each $a\in A$, let $f_a \in S$. Show that $\sup \{ f_a:a\in A \} \in S$.
Some classmates and I are confused as to what the supremum is doing here. If the supremum is just one $f_a$, for instance say $f_k$ for some $k \in A$ then it is trivial. It must mean something else. I think we are just tripped up on the meaning of the supremum of a sequence of functions.
Edit:
After clarification it must mean supremum regarding $x$.