On Topoiogical Spaces with A σ-Almost Locally Finite Base

Abstract:

      In 1983, ML Ito and K.Tamano introduced the notion of almost locally finiteness and investigated some properties of the class of spaces with a σ almost locally finite base. In this paper, some more results about such spaces are given . The main results are as follows:( 1 ) The image of a space with a σ -almost locally finite base under the finite to one closed map has a σ-almost locally finite base;( 2 ) The locally finite sum theorem holds for such spaces;( 3 ) A space X has a σ almost locally finite base if and only if X is a paracompact σ space and every closed subset of X has an almost locally finite open neighborhood base; (4)The monotonically normal space with an M-structure has a a σalmost localiy finite base.