Small spaces which “generate” large spaces

Abstract:

Let $\underline S$ denote the class of Tychonoff spaces. For $X \in \underline S$ and $\underline C \subseteq \underline S$, we say that $X$ generates large $\underline C$-spaces if for every cardinal $\alpha$ there is $Y \in \underline S$ such that $Y \in \underline C, X \subseteq Y$, and every $Z \in \underline {C}$ with $X \subseteq Z \subseteq Y$ satisfies $|Z| > \alpha$. For classes $\underline C$ which satisfy certain mild and natural conditions, we show for each $X \in \underline S$ that $X$ generates large $\underline C$-spaces iff there is no weakly free $\underline C$-space over $X$—i.e., no space $Y$ such that $X \subseteq Y \subseteq C$ and every continuous $f:X \to Z \in \underline C$ extends to a continuous function $\bar f:Y \to Z$. Among the classes $\underline C \subseteq \underline S$ which satisfy these conditions for every $X \notin \underline C$ are the class of pseudocompact Tychonoff spaces and the class of almost compact (= absolutely ${C^*}$-embedded) Tychonoff spaces.