A Study On Regular ?~n-Paracompact Spaces,?~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(?) ={f?:???)be a family of generalized continuous maps,where f? X ? Y?is a mapping from the generalized topological space.X to the generalized topological space Y? for each ??A,if (?) separates points and separates points and generalized closedsets,then f?X ??Y?,where,f(x)=<f?(x):???)for each x?X and ? Y? 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.