Publisher : Department of Mathematics, Kyungpook National University
DOI : 10.5666/KMJ.2010.50.4.537
Title & Authors
On a Question of Closed Maps of S. Lin Chen, Huaipeng;
Let X be a regular -space such that each single point set is a set. Denot 'hereditarily closure-preserving' by 'HCP'. To consider a question of closed maps of S. Lin in , we improve some results of Foged in , and prove the following propositions. Proposition 1. is discrete and closed if is a collection of HCP. Proposition 2. is an fininte subcolletion of is HCP if is a collection of HCP. Proposition 3. Let (X,) have a -HCP k-network. Then (X,) has a -HCP k-network F = such that such tat: (i) , (ii) is a discrete closed set and (iii) each is closed to finite intersections.
L. Foged, A characterization of closed images of metric spaces, Proc. Amer. Math. Soc., 95(1985), 487-490
G. Gruenhage, General metric spaces and metrization, in: M. Husek and J. Van Mill, Editors, Recent Progress in General topology, Chapter 7 240-274.
G. Gruenhage. Generalized metric spaces, in: K. Kunen and J. E. Vaughan, Eds., handbook of Set-Theoretic Topology 423-501.
N. Lasnev, Continuous decompositions and closed mappings of metric spaces, Sov. Math. Dokl., 165(1965), 756-758.
N. Lasnev, Closed mappings of metric spaces, Sov. Math. Dokl., 170(1966), 505-507.
S. Lin, A survey of the theory of $\aleph-spaces$, Q and A in Gen. Top., 8(1990), 405-419.
E. Michael, A note on closed maps and compact sets, Israke J. Math., 2(1964), 173-176.
E. Michael, A quintuple quotient quest, Gen. Topology and Appl., 2(1972), 91-138.