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Exponential domination in function spaces

Vladimir Vladimirovich Tkachuk (2020)

Commentationes Mathematicae Universitatis Carolinae

Given a Tychonoff space X and an infinite cardinal κ , we prove that exponential κ -domination in X is equivalent to exponential κ -cofinality of C p ( X ) . On the other hand, exponential κ -cofinality of X is equivalent to exponential κ -domination in C p ( X ) . We show that every exponentially κ -cofinal space X has a κ + -small diagonal; besides, if X is κ -stable, then n w ( X ) κ . In particular, any compact exponentially κ -cofinal space has weight not exceeding κ . We also establish that any exponentially κ -cofinal space X with...

Exponential separability is preserved by some products

Vladimir Vladimirovich Tkachuk (2022)

Commentationes Mathematicae Universitatis Carolinae

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 σ -compact crowded space in which all countable subspaces are scattered. If X is a Lindelöf space and every Y X with | Y | 2 ω 1 is scattered, then X is functionally countable; if every Y X with | Y | 2 𝔠 is scattered, then...

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