| Since Heath [1962] introduced a set-valued function - g-functions, he not only used g-function successfully to answer the problems of McAuley and Brow ,but also proved the characterizations of stratifiable spaces and -spaces in terms of g-functions.Moreover , he showed that the stratifiable spaces are a-spaces which answered the question of ArhangePskii[1966]. These works showed up initially the power of g-functions.Moreover ,Na-gata[l] Y.ziqiu[12] pointed out ,g(n, x) is only a neighborhood (not necessarily open) of z in most cases; R.E.Hodel[4] introduced -structure:R.E.Hodel[4] called {g(n,x)|x X, n N] of X is an -structure,if:E g(n,x) for all n N andx X; they also gave some good results in terms of g-function and -structure.Since then, Heath, Hodel, Fletcher, Lindgren and Nagata showed that the g-function is a powerful tools for studying the generalized metric spaces.Junnila and Yajima introduced several subclasses of the classes of -spaces. The smallest of the classes considered,that of LF-netted spaces ,contains all F -discrete spaces and all stratifiable F -metrizable spaces.The main result of the paper establishes the equivalence of normality and countable paracompactness of the product of an LF-netted space with a count-ably paracompact and normal space.In chapter 1 ,we introduce background of the g-functions and LF-netted spaces;In chapter 2 ,we give several results concerning g-function and -structure;.It contains the following results :Theorem 2.1.2 A regular space X is mertizible X has a -LFnetwork F = Fn such that each Fn is pip.Theorem 2.1.4 If A regular space X has a -cp network F = Fnsuch that each Fn is pip ,then it has a g-function satisfying: (i)if x 6 g(n, xn), then xn - x; (ii)for each A C X and n e N,A {g(n,y)|y 4}. Theorem 2.1.6 A regular Frechet space is Lasnev there is a g-function satisfying:(ii) n is hep for eath n.Theorem 2.1.9 A space X is -space There exists a function g : N X - r such that:(a)y y(n, a;) = p(n, y) g(n, x) for each n N; (c){g(n, y)|y e p(n, x)} is a finite collection for each x X andn N.(d) n is a locally finite collection for each n N.Theorem 2.2.2 A regular Frechet space X is N if and only if it has a -structure g which satisfies the following two conditions:(i)xn - x, xn g(n, yn) = yn - x; ;and(ii) {g(n, x)|x X} is a locally-finite collections for each n N.Theorem 2.2.4 A regular space is LF-netted(or HCP-netted) iff there is an -structure {g(n,x)| x X, n N} that satisfies conditions:{g(n,x)|g(n, x) S 0} is a locally-finite (or hereditarily closure-preserving) collection in the set X \ S for every closed subset S of X;(iv){g(n, x)|x X} is a locally-finite (or hereditarily closure-preserving) collection for each n N.In chapter 3, we study the P-netted spaces and a subclasses of the classof LF-netted spaces,and consider the relations of LF-k-netted spaces with several generalized metrizable spaces.we also give the equivalence of LF-k-netted spaces.The main results are:F -metrizable is LF-netted, F-metrizable space is -space moreover, monotonically normal -spaces is stratifiable,so monotonically normal F-metrizable spaces is LF-netted ; in terms of [1,Theorem 4] and [25],we may get the following theorem :Theorem 3.1.2 If a regular Frechet space has a network T = n=1 Fn such that each Fn is a locally finite closed cover of X ,and such that for each x X and Fn with x Fn Fn for all n N,{Fn|n N} is either a network at x or HCP, then X is LF-netted.Theorem 3.1.4 If F = n=1 Fn is a -CP collection of X,H = Um=1 Hm is -CP collection of Y .Assmue F H is CP-regular in X x Y,then X x Y is CP-netted.Example 3.1.5 There exists a semimetrizable space which is not CP-netted.Example 3.2.5 There exists a LF-netted space which is not a LF-k-space.Theorem 3.2.6 Every metric space is LF-k-netted.Proposition 3.2.9 :(A)The image of an LF-k-netted space under a perfect map is LF-k-netted.(B)The continuous image of a CP-k-netted(HCP-k-netted) space under a close... |
PDF Full Text Request