Mapping theorems on spaces with sn-network g-functions

Let Δ be the sets of all topological spaces satisfying the following conditions. (1) Each compact subset of X is metrizable; (2) There exists an sn-network g-function g on X such that if xn → x and yn Є g(n, xn) for all n Є N, then x is a cluster point of {yn}. In this paper, we prove that if X Є Δ, then each sequentially-quotient boundary-compact map on X is pseudo-sequence-covering; if X Є Δ and X has a point-countable sn-network, then each sequence-covering boundary-compact map on X is 1-sequence-covering. As the applications, we give that each sequentially-quotient boundary-compact map on g-metrizable spaces is pseudo-sequence-covering, and each sequence-covering boundary-compact on g-metrizable spaces is 1-sequence-covering.