The concepts of -systems, -networks and -covers were defined by A. Arhangel’skiǐ in 1964, P. O’Meara in 1971 and R. McCoy, I. Ntantu in 1985, respectively. In this paper the relationships among -systems, -networks and -covers are further discussed and are established by -systems. As applications, some new characterizations of quotients or closed images of locally compact metric spaces are given by means of -systems.
For a Tychonoff space , let be the family of hypographs of all continuous maps from to endowed with the Fell topology. It is proved that has a dense separable metrizable locally compact open subset if is metrizable. Moreover, for a first-countable space , is metrizable if and only if itself is a locally compact separable metrizable space. There exists a Tychonoff space such that is metrizable but is not first-countable.
For any class 𝒦 of compacta and any compactum X we say that: (a) X is confluently 𝒦-representable if X is homeomorphic to the inverse limit of an inverse sequence of members of 𝒦 with confluent bonding mappings, and (b) X is confluently 𝒦-like provided that X admits, for every ε >0, a confluent ε-mapping onto a member of 𝒦. The symbol 𝕃ℂ stands for the class of all locally connected compacta. It is proved in this paper that for each compactum X and each family 𝒦 of graphs, X is confluently...