Weakly quasi-first-countable spaces and box products

A space X is said to be weakly quasi-first-countable if and only if for all x ? 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 ? O, O contains a member of each family associated to x. For a space X, we denote the countable ?-product of X endowed with the box topology by ?B(X). We prove that if X is first-countable and locally compact, then ?B(X) is weakly quasi-first-countable, which gives a general method to construct weakly quasi-firstcountable spaces which are neither weakly first-countable nor quasi-first-countable.