Embedding S(X) into S(Y) when Y is compact and S is not.
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...