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May 4, 2020 at 17:55 comment added Tim Campion In fact -- doesn't the usual counterexample to cartesian closure of $Top$ just use the quotient of $\mathbb R$ by some countable subset? If I understand correctly, this construction should be perform-able in your category.
May 4, 2020 at 17:53 comment added Tim Campion I doubt it -- the usual way to show cartesian closure is to use that the category is generated under colimits by exponentiable spaces, which won't be the case here.
Apr 28, 2020 at 15:08 history edited Martin Sleziak
added the (cartesian-closed-categories) tag
Apr 28, 2020 at 15:02 history asked Rupert CC BY-SA 4.0