It follows from Lawvere's theorem that for most spaces $X$ there is no space-filling curve for its path space, $\alpha: I \to X^I$, working here in the category of $k$-spaces. (Yes, that would also follow where one knows $X^I$ is not compact, but one point is that a similar result holds replacing $I$ by $\mathbb{R}$).

The category of Polish spaces is not cartesian closed, because every Polish space $B$ admits a continuous surjection $B\to X$ from Baire space $B$ (the space of irrationals), so that in particular there is no Polish function space $X^B$ for most Polish spaces $X$.

It follows from Lawvere's theorem that for most spaces $X$ there is no space-filling curve for its path space, $\alpha: I \to X^I$, working here in the category of $k$-spaces. (Yes, that would also follow where one knows $X^I$ is not compact, but one point is that a similar result holds replacing $I$ by $\mathbb{R}$).

The category of Polish spaces is not cartesian closed, because every Polish space $X$ admits a continuous surjection $B\to X$ from Baire space $B$ (the space of irrationals), so that in particular there is no Polish function space $X^B$ for most Polish spaces $X$.

It follows from Lawvere's theorem that for most spaces $X$ there is no space-filling curve for its path space, $\alpha: I \to X^I$, working here in the category of $k$-spaces.

It follows from Lawvere's theorem that for most spaces $X$ there is no space-filling curve for its path space, $\alpha: I \to X^I$, working here in the category of $k$-spaces. (Yes, that would also follow where one knows $X^I$ is not compact, but one point is that a similar result holds replacing $I$ by $\mathbb{R}$).

The category of Polish spaces is not cartesian closed, because every Polish space $B$ admits a continuous surjection $B\to X$ from Baire space $B$ (the space of irrationals), so that in particular there is no Polish function space $X^B$ for most Polish spaces $X$.