Sequence-covering maps on generalized metric spaces

Abstract

Let $f:X\to Y$ be a map. $f$ is a {\it sequence-covering map}\cite{Si1} if whenever $\{y_n\}$ is a convergent sequence in $Y$ there is a convergent sequence $\{x_n\}$ in $X$ with each $x_n\in f^{-1}(y_n)$; $f$ is an {\it 1-sequence-covering map}\cite{Ls2} if for each $y\in Y$ there is $x\in f^{-1}(y)$ such that whenever $\{y_n\}$ is a sequence converging to $y$ in $Y$ there is a sequence $\{x_n\}$ converging to $x$ in $X$ with each $x_n\in f^{-1}(y_n)$. In this paper, we mainly discuss the sequence-covering maps on generalized metric spaces, and give an affirmative answer for a question in \cite{LL1} and some related questions, which improve some results in \cite{LL1, Ls4, YP}, respectively. Moreover, we also prove that open and closed maps preserve strongly monotonically monolithity, and closed sequence-covering maps preserve spaces with a $\sigma$-point-discrete $k$-network. Some questions about sequence-covering maps on generalized metric spaces are posed.

13 pages