A Study On Regular Δ~n-Paracompact Spaces,Δ~n-Normal Spaces And Generalized Compactifications

In this thesis,we introduce concepts of regulrdr△n-paracompact,△n-normal and△n-metacompact. We get some properties of them. We also introduce a notion of a generalized compactification of a generalized topological space.We get some conclusions on generalized compactifications.Fix a topological space X,let n∈N,n≥2.Let△nZ={<x,x,…,x>∈Xn:x∈X}. X is regular △n -paracompact if for every C (?) Xn\△nX closed in Xn there exists a locally finite open cover u of X such that ∪{u-:u∈u}does not meet C.X is△n -normal if for every X (?) Xn\△nX closed in Xn there exist disjoint open sets U and V of Xn such that C (?)U and△nX (?) V.X is△n -metacompact if for every C (?) Xn\△nX closed in Xn there exists a point-finite open cover u of X such that U{Un:U∈u)does not meet C.Firstly,we show that a space X is regular△n -paracompact if and only if for every open cover 9 of X,there is a locally finite open cover u of X such that for each M (?) U- with U∈u and |M|≤n,there is a G∈(?) with M (?) G.We show that a space X is△" normal if and only if for every open cover 9,there exists an open cover u of X such that ClXn(U{Un:U∈u))(?) U{Gn:G∈(?) ).We also show that a space X is△n -metacompact if and only if for every open cover 9 of X,there is a point-finite open cover u of X such that for each M (?) U with U∈u and |M|≤n,there is a G∈(?) with M (?) G.Secondly,we get that a normal△n -paracompact space is regular△n -paracompact,where n∈N and n≥2. We get that a△n -normal and△n -paracompact space is regular△n -paracompact for each n ∈ N. We get some properties of regular△n -paracompact spaces. We show that a regular△ -paracompact and orthocompact space is regular△n -paracompact for n≥2.Y Hirata posed the following problem.Let X be a △ -paracompact and orthocompact space.Is.X △2+2/1 -paracompact? We know that a metacompact space is orthocompact.We show that every △ -paracompact and metacompact space is △ 2+2/1 -paracompact.Finally,we mainly study a generalized compactification of a generalized topological space. We get some properties on generalized continuous maps and generalized open(closed)maps. We show that let(?) ={fα:α∈Λ)be a family of generalized continuous maps,where fα X → Yαis a mapping from the generalized topological space.X to the generalized topological space Yα for each α∈A,if (?) separates points and separates points and generalized closedsets,then f：X →ПYα,where,f(x)=<fα(x):α∈Λ)for each x∈X and П Yα is a generalized product space,is a generalized embedding.We also show that if.X be a generalized completely regular space,there exists a generalized compactification Y of X having the property that every bounded generalized continuous map f:X→R extends to a generalized continuous map of Y into R.