Volume 37 • Number 3 • 2014   • Weakly Quasi-First-Countable Spaces and Box Products Rongxin Shen and Fucai Lin Abstract. A space $X$ is said to be weakly quasi-first-countable if and only if for all $x \in X$, there exists countably many countable families of decreasing subsets containing $x$ such that a set $O$ is open if and only if for any $x\in O$, $O$ contains a member of each family associated to $x$. For a space $X$, we denote the countable $\sigma$-product of $X$ endowed with the box topology by $\sigma B(X)$. We prove that if $X$ is first-countable and locally compact, then $\sigma B(X)$ is weakly quasi-first-countable, which gives a general method to construct weakly quasi-first-countable spaces which are neither weakly first-countable nor quasi-first-countable. 2010 Mathematics Subject Classification: 54A05, 54D15 Full text: PDF