Discontinuous uncurrying map

Question

Suppose $f: X\to \mathcal{C}(Y,Z)$ is a continuous map of topological spaces. The uncurrying map is $\beta_f: X\times Y\to Z$, $(x,y)\mapsto f(x)(y)$. I know that this $\beta_f$ map is always continuous if $Y$ is locally compact and Hausdorff. So this led me to try to find an example of such an $f$ which leads to a discontinuous $\beta_f$. My thought was to use $\mathbb{Q}=Y$ (with subspace topology) since this space is not locally compact. But I could not find an appropriate $f$ that makes $\beta_f$ not continuous.

Any thoughts?

Edit: Note $\mathcal{C}(Y,Z)$ has the compact open topology. Ie, the topology generated by $o(K,U)$ where $K\subset Y$ is compact and $U\subset Z$ is open.

I think you'd be better off trying not Hausdorff. I'm thinking the indiscrete topology. — Matt Samuel, Nov 30 '19 at 19:16
The only continuous functions in the indiscrete topology are constant. It's just a guess, I don't know if it would work out. — Matt Samuel, Nov 30 '19 at 20:13
What topology are you putting on $\mathcal{C}(Y,Z)$? The compact-open topology? — Eric Wofsey, Nov 30 '19 at 20:22

score 1 · Answer 1 · answered Dec 01 '19 at 00:53

This is essentially a duplicate of $\operatorname{Hom}(X \times Z, Y) \cong \operatorname{Hom}(X, \operatorname{Map}(Z, Y))$ is not true in $\textbf{Top}$

Let $X= C(Y,Z)$ and $f = id$. Then $\beta_f$ is the evaluation map $\Omega : C(Y,Z) \times Y \to Z,(\phi,y) \mapsto \phi(y)$.

Now see $\operatorname{Hom}(X \times Z, Y) \cong \operatorname{Hom}(X, \operatorname{Map}(Z, Y))$ is not true in $\textbf{Top}$ for examples where $\Omega$ is not continuous. In fact, $Y = \mathbb Q$ and $Z = [0,1]$ will do.

Discontinuous uncurrying map

1 Answers1