Some new results on $\hat{I}”$-spaces

Abstract

A topological space $X$ is a $\hat{I}”$-space (or $X\in \hat{I}”$) if for any decreasing sequence $\{A_n:n<\ddot{I}‰\}$ of subsets of $X$ with empty intersection there is a (decreasing) sequence $\{U_n:n<\ddot{I}‰\}$ of open sets with empty intersection such that $A_n\subset U_n$ for all $n<\ddot{I}‰$. In this note we prove the following results concerning $\hat{I}”$-spaces. 1) Every $T_3$ countably compact $\hat{I}”$-space is compact. 2) If there is a $T_1$ crowded Baire $\hat{I}”$-space then there is an inner model with a measurable cardinal. 3) If $X\in \hat{I}”$ and $cf(o(X))>\ddot{I}‰$ then $|X|<o(X)$. (Here $o(X)$ is the number of open subsets of $X$.) The first two of these provide full and/or partial solutions to problems raised in the literature, while the third improves a known result.

8 pages