2
votes
0answers
78 views

How are measurable functions morphisms?

I am trying to encode the theory of measurable sets in higher order logic. I already did so with the theory of topology. I think it is relevant because it enables one to see continuous or measurable ...
7
votes
3answers
708 views

What is the definition of continuity of set-valued functions?

According to the wiki of Kakutani's fixed-point theorem, A set-valued mapping $\varphi$ from a topological space $X$ into a powerset $\wp(Y)$ called upper semi-continuous if for every open set $W ...
3
votes
1answer
219 views

Automatic continuity of the inverse map

All topological spaces considered here are Hausdorff. It is a well-known consequence of the minimality of a compact topology that an injective continuous map $f\colon X\to Y$ where $X$ is compact, ...
2
votes
1answer
205 views

Function spaces over pseudocompact spaces

Let $K$ be a pseudocompact Tychonoff space that is, a Hausdorff $T_{3.5}$ space for which every continuous function $\varphi \colon K\to \mathbb{C}$ is bounded. Let $\beta$ be the Stone-Cech extension ...
5
votes
0answers
197 views

Topological conditions forcing continuity

Let $X$, $Y$, and $Z$ be topological spaces. Let $f:X \rightarrow Y$. Further assume that for every continuous function $g:Y \rightarrow Z$, $g \circ f$ is continuous. Question: Under what ...
0
votes
4answers
2k views

Does Cauchy continuity imply uniform continuity? [No.] [on hold]

It is well known that if $X$ is a first countable topological space and $Y$ is a topological space, then $f : X \rightarrow Y$ is continuous iff $$\forall x \in {\rm map}(\mathbb{N},X),\forall p \in X ...