Exponentiability in homotopy slices of TOP and pseudo-slices of CAT.
Given a Tychonoff space and an infinite cardinal , we prove that exponential -domination in is equivalent to exponential -cofinality of . On the other hand, exponential -cofinality of is equivalent to exponential -domination in . We show that every exponentially -cofinal space has a -small diagonal; besides, if is -stable, then . In particular, any compact exponentially -cofinal space has weight not exceeding . We also establish that any exponentially -cofinal space with...
The purpose of this note is to prove the exponential law for uniformly continuous proper maps.
We show that exponential separability is an inverse invariant of closed maps with countably compact exponentially separable fibers. This implies that it is preserved by products with a scattered compact factor and in the products of sequential countably compact spaces. We also provide an example of a -compact crowded space in which all countable subspaces are scattered. If is a Lindelöf space and every with is scattered, then is functionally countable; if every with is scattered, then...