半拓扑线性空间及其性质(Ⅰ)

Semi-topological Linear Spaces and Its Properties(Ⅰ)

首先在一般的拓扑空间中引入了准半连续映射的概念,并借助此概念引入了半拓扑线性空间,得到了这一新空间的如下基本性质:(1)给出了半拓扑线性空间中半开集和0点的邻域的特征刻画;证明了半拓扑线性空间中0点的局部S基可通过平移作为任何一点的局部S基;证明了半拓扑线性空间中半开集和任何集的和仍然是半开集.(2)证明了半拓扑线性空间的局部S基的每一个元是吸收集,并且它包含0点的一个平衡S邻域;证明了对具有C性质的半拓扑线性空间中的0的每个S邻域u,存在0点的S邻域v,使得v的半闭包v-u;证明了对具有C性质的半拓扑线性空间的局部S基的每一个元u,存在局部S基的元v,使v+vu成立.(3)给出了半拓扑线性空间中有关半闭包和半内部的等式或蕴涵关系.

In this paper,we introduce the notions of pre-semi-continuous mapping in general topological spaces and semi-topological linear spaces by using the notion of pre-semi-eontinuotius mapping,and we obtain the some fundamental properties of semi-topological linear spaces： （1）the properties of S-heighhborhood are given in semi-topological linear spaces and proved the local S-base of any points were transfer by local S-base of zero point in semi-topological linear spaces, also, semi-open sets added any sets are semi-open sets were proved; （2）any elements of local S -base are absorbing sets and containt a balanced sets of zero point, each S -heighhborhood u of zero point in semi-topological linear spaces with C -properties exsits S -heighhborhood v of zero point , supposing it v- is semi-closure sets, then v_ belong to u , also, for each elements of local S -base u in semi-topological linear spaces with C -properties , there is an element v of local S -base, made v ＋ v belong to u ; （3） the equality of semi-closure and semi-interion sets in semi-topological linear spaces are given.