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Stream: deprecated: analysis

Topic: A fibrewise extreme value theorem

Graham Manuell (Jan 23 2023 at 17:59):

Let $p\colon X \to B$ be a continuous surjection which is both open and proper and let $f\colon X \to \mathbb{R}$ be continuous function. Define a function $g\colon B \to \mathbb{R}$ so that $g(b)$ is the maximum value that $f$ takes when restricted to the fibre $p^{-1}(\{b\})$. Then (at least under the assumption of a mild separation axiom) the map $g$ is continuous.

Has anyone heard of this result before and/or does anyone find this result obvious? Alternatively, do you know a resource that might mention this result. It is not in Fibrewise Topology by James and I haven't managed to find anything online.

Graham Manuell (Jan 23 2023 at 18:06):

I care because I am using this as an easy example of a consequence of the constructive extreme value theorem applied to a sheaf topos and it would be nice if the result was not dead obvious to someone with any experience in fibrewise topology and/or if it had an application somewhere.

Simon Willerton (Jan 23 2023 at 23:01):

It reminds me of the stuff in Roald Koudenburg's paper A categorical approach to the maximum theorem. The result that Roald is talking about he refers to as "Berge's maximum theorem". I don't know if that's helpful.

Graham Manuell (Jan 23 2023 at 23:18):

Ah. That does seem relevant! Thanks!

Matteo Capucci (he/him) (Jan 25 2023 at 14:44):

Simon Willerton said:

It reminds me of the stuff in Roald Koudenburg's paper A categorical approach to the maximum theorem. The result that Roald is talking about he refers to as "Berge's maximum theorem". I don't know if that's helpful.

Nice reference!