Compact mappings and s-mappings at subsets

Abstract

Abstract Almost s s -mappings and almost compact mappings have been introduced and studied. In this article, we continue to research some questions related to the almost s-images (resp., almost compact images) of metric spaces. The following results are obtained. (1) A space X X is a quotient and almost compact image of a metric space if and only if X X is a sequential space having a c s ∗ c{s}^{\ast } -network which is point-regular at nonisolated points, which gives an affirmative answer to Question 4.9 in the article “S. Lin, X. W. Ling, and Y. Ge, Point-regular covers and sequence-covering compact mappings, Topology Appl. 271 (2020), 106987.” (2) There exists a bi-quotient and almost compact image of a metric space satisfying no base, which is point-countable at nonisolated points, which gives negative answers to Question 3.1 in the article “X. W. Ling and S. Lin, On open almost s-images of metric spaces, Adv. Math. (China) 48 (2019), no. 4, 489–496” and Question 3.7 in the article “X. W. Ling, S. Lin, and W. He, Point-countable covers and sequence-covering s-mappings at subsets, Topology Appl. 290 (2021), 107572.” (3) Some characterizations of countably bi-quotient and almost s s -images (resp., pseudo-open and almost compact images) of metric spaces.