5
$\begingroup$

A topological space $X$ is exponentiable, meaning that that an exponential object $Y^X$ exists for every topological space $Y$, iff $X$ is core-compact.

However, in the only proof I know of that non-core-compact spaces are non-exponentiable (here: http://www.cs.bham.ac.uk/~mhe/papers/newyork.pdf), the counterexample given is that there is no exponential object $\mathbb{S}^X$, where $\mathbb{S}$ is Sierpinski space, which is a kind of strange topological space.

Are there known theorems of the form "If (and possibly only if) $X$ satisfies some property P (weaker than core-compactness), then an exponential object $Y^X$ exists for every space $Y$ in some class $\mathcal{C}$ of nice topological spaces"? Alternatively, is it known that even for certain narrow classes $\mathcal{C}$ of topological spaces, if an exponential object $Y^X$ exists for every $Y\in\mathcal{C}$, then $X$ must be core-compact?  

Improve this question
$\endgroup$
1
  • 5
    $\begingroup$ Well, for instance, if you require $Y$ to be Hausdorff, then $Y^X$ exists for any space $X$ with no non-constant maps to a Hausdorff space (namely, $Y^X=Y$). So if you are imposing some niceness conditions on $Y$, you will probably want to also impose conditions on $X$ saying that $X$ is nice enough to be able to have nontrivial maps to $Y$ in order to avoid trivialities. That's why the Sierpinski space is a natural choice of $Y$ here: every space has "enough" maps to it. $\endgroup$
    – Eric Wofsey
    Commented May 3, 2017 at 1:22

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .