Abstract
We establish that a Tychonoff space X is countable if and only if Cp(X) is strongly dominated by a second countable space. The same is true for a compact space K such that Cp(K,[0,1]) is strongly dominated by a second countable space. We also prove that strong domination by a second countable space of the complement of the diagonal of a Tychonoff space X implies that X is an ℵ0-space. Our results solve several published open questions.
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