The P-covering Images Of Metric Spaces And The Closed Inverse Images Of Weak Submesocompact Space

This paper investigates the P-covering images of metric spaces and the closed inverse images of weak submesocompact space.In the first part, this paper proves that if U is a point-countable family of subset of a space X, then each subset B of X has at most countably many minimal cfp-covers by elements of U, which improves a result of S. Lin and P. Yan by omitting compactness of B. As an application of this result, this paper proves that a space X is a P-covering, s-image of a metric space if and only if X has a point-countable Pfp-network, where P is a topological property implying countable compactness and satisfying closed heredity.In the second part, this paper investigates inverse preservation of weak submeso-compactness (a natural generalization of submesocompactness) under mappings. This paper proves perfect mappings inversely preserve weak submesocompactness and closed Lindelof mappings inversely preserve weak submesocompactness if domains and images are regular.