A characterization of Lašnev spaces

Abstract

ABSTRACT. We give here a characterization of closed images of metrizable spaces in terms of the primitive concept of an ever finer sequence of partitions and a requirement considerably weaker than that of a fc-network.Almost twenty years after Lasnev's [2] first internal characterization of closed images of metrizable spaces came Foged's [1] improvement. Foged made use of the notion of Ac-networks of Michael [3] and came up with a much neater result. Nagata [5,6], inspired by his student Sasaki's work, gave another solution in terms of gfunctions. In all these three works, unmistakable is the motif of a cr-hereditarily closure-preserving (cr-HCP) collection of subsets. We give here a characterization of Lasnev spaces, as these closed images of metrizable spaces have come to be called, in terms of the much more primitive concept of an ever finer sequence of partitions and a requirement considerably weaker than that of a fc-network, which, when in conjunction, one with the other, nonetheless become at once a cr-HCP ¿-network of subsets, the range of a ç/-function, into the set of all subsets instead of the topology, but otherwise of Nagata [5,6] and an A-system of cr-HCP coverings the like of which we first encountered in Lasnev [2].