In this paper, we prove that each sequence-covering and boundary-compact map on $g$-metrizable spaces is 1-sequence-covering. Then, we give some relationships between sequence-covering maps and 1-sequence-covering maps or weak-open maps, and give an affirmative answer to the problem posed by F.C. Lin and S. Lin in [].

In this paper, we give an affirmative answer to the problem posed by Y. Tanaka and Y. Ge (2006) in "Around quotient compact images of metric spaces, and symmetric spaces", Houston J. Math. 32 (2006) no. 1, 99-117.

Some relationships between $1$-sequence-covering maps and weak-open maps or sequence-covering $s$-maps are discussed. These results are used to generalize a result from Lin S., Yan P., , Topology Appl. (2001), 301–314.

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