Maintain A Family Of Sets And Tight Finite Family Of Sets On The Closure Of Weak Genetic Research

Since 1960s, the investigations of generalized metric spaces is an active direction of General Topology all the time. Because the structure of networks is more delicate and more variable, the topology scholars have made all sorts of restricts to various networks. Then many classes of generalized metric spaces were draw in and studied.Topology scholars have investigated spaces with σ—discrete, σ— locally finite, σ — HCP various networks, and the relationships between them.In this thesis, we investigate the relationships between six kinds of spaces with σ—ωHCP, σ—compact-finite k-networks, cs-networks and wcs~*-networks.In addition, we develop the theory of generalized metric spaces by these relations. The primary studies in this paper are the following:(1)We obtain the relationships between spaces with σ—ωHCP k-networks (cs-networks, wcs~*-networks) and with σ—compact-finite k-networks (cs-networks, wcs~*-networks). And the relationships between spaces with σ—ωHCP (σ—compact-finite) k-networks, cs-networks, wcs~*-networks respectively.(2)We apply the above relations to develop the theory of generalized metric spaces. We also obtain a new judgement theorem for hereditarily metaLindelof space and paracompact space. In addition, we give a depiction for g-metrizable space from a new direction with σ—ωHCP closed weak base. They enrich the basic theory of generalized metric spaces.In addition, through summary of results in this thesis, we can...