Stratifiable spaces defined by pair collections

AbstractWe define a pair (F,U) to be a closed set F and an open set U such that F ⊂ U. A sequence of pair collections is used to characterize stratifiable spaces instead of a sequence of neighbornets. We introduce a new class of spaces, called regularly stratifiable spaces, which is defined in terms of pair collections. Every stratifiable μ -space is regularly stratifiable, and every regularly stratifiable space has a σ -almost locally finite base, thus is hereditary M1. J. Nagata's problem for the dimension of M1 -spaces is answered positively in the class of regularly stratifiable spaces