Sn-metrizable Spaces and Related Matters

sn-networks were first introduced by Lin [12], which are the concept between weak bases and cs-networks. sn-metrizable spaces [6] (i.e., spaces with σ-locally finite sn-networks) are one class of generalized metric spaces, and they play an important role in metrization theory, see [6, 13]. In this paper, we give a mapping theorem on sn-metrizable spaces, discuss relationships among spaces with point-countable sn-networks, spaces with uniform sn-networks, spaces with locally countable sn-networks, spaces with σ-locally countable sn-networks, and sn-metrizable spaces, and obtain some related results. In this paper, all spaces are regular and T1, all mappings are continuous and surjective. N denotes the set of all natural numbers. ω denotes N∪{0}. For a family of subsets of a space X and x ∈ X , denote ( )x = {P ∈ : x ∈ P}. For two families and of subsets of X , denote ∧ = {A∩B : A∈ and B ∈ }.