A Study On Topological Function Spaces,Semitopological Groups And Paratopological Groups

Topological algebra is a major research direction in general topology,and topological function spaces are closely related topological algebra.It is well known that if Y is a topological group,then so is Cp(X,Y),the space of continuous functions from X to Y with the topology of pointwise convergence.In recen-t years,topologists pay special attention to the study of properties of objects such as Cp(X),Cp(X,n),topological groups,paratopological groups,semitopo-logical groups,quasitopological groups in topological algebra.This thesis mainly investigates the relationship between the Dedekind remainder of GO-spaces and spaces of continuous step functions over them,the Hausdorff reflection and three space properties on semitopological groups,and subgroups of products of certain paratopological or semitopological groups.First,properties of spaces of continuous step functions Cp(L,n)over a GO-space L are investigated.Given a GO-space L,the Dedekind completion of L is denoted by cL and the subspace of Cp(L,n)consisting of constant functions and step functions with only finitely many steps is denoted by Sp(L,n).An element x ? cL is in T(L)if and only if x ? cL\L,or x=?,or x? L and x has the immediate successor in L.Points of T(L)that are in L are declared isolated.The other points inherit base neighborhoods from the Dedekind completion cL.We show that if L is a GO-space then T(L)n is covered by finitely many closed homeomorphic copies of a closed subspace of Cp(L,n+1),and then we show that if L is a GO-space and T(L)n is Lindelof(Menger)for each n then Sp(L,n)is Lindelof(Menger)for each n;if L is a countably compact GO-space then Cp(L,n)is Menger for each n if and only if T(L)is Lindelof;if L is a first countable GO-space such that L.={x? L:x is not isolated} is countably compact and Y=L\L'? L' is scattered with rank(Y)<?1,then Cp(L,m)is a Menger space if and only if Cp(L,m)is a Lindelof space for m?N.Besides,we investigate the relationship between Cp(L,n)and Sp(L,n)and show that if L is a GO-space then Sp(L,n)is dense in Cp(L,n)for each n? N.Second,we show that many topological properties are invariant or inverse invariant under taking T2 reflections in semitopological groups and extend some three space properties in topological groups(paratopological groups)to semitopo-logical groups,establishing the following result:Let G be a regular semitopological group and let H be a closed subgroup of G such that all compact(resp.,count-ably compact,sequentially compact)subsets of the semitopological group H are first-countable;If the quotient space G/H has the property that all compact(re-sp.,countably compact,sequentially compact)subsets are Hausdorff and strongly Frechet(strictly Freechet),then so does the semitopological group G.Finally,we investigate properties of subgroups of products of certain paratopological(semitopological)groups:We give some sufficient conditions un-der which a paratopological group is topologically isomorphic to a subgroup of a product of strongly metrizable paratopological groups,and show that a regular(Hausdorff,T1)semitopological group G admits a topologically isomorphic em-bedding as a subgroup into a product of regular(Hausdorff,T1)first-countable semitopological groups which are ?-spaces if and only if G is locally ?-good,?-balanced,Ir(G)??(Hs(G)??,Sm(G)??)and with the property that for every open neighborhood U of the identity e of G the cover {xU:x? G} has a basic refinement F which is ?-discrete with respect to a countable family V of open neighborhoods of e,and then we give an internal characterization of projectively Ti second-countable semitopological groups for i=0,1,2,which give an answer to Problem 3.1 in[132]in the cases of i=0,1,2.