The Boundary Compact Image Of Metric Spaces

This paper mainly discuss the spaces under the boundary compact image of metric spaces.It is shown that each sequence-covering and boundary compact map on a metric space is 1-sequence-covering.It is also studied the image of sequentially quotient(sequence-covering),boundary compact image of a metric space,and obtain the characeters of space with a point countable sn-network and snf-countable space. The concept of space with an uniform base at non-isolated points is introduced.It is shown that X is an open,boundary-compact image of a metric space if and only if X has an uniform base at non-isolated points.Some relationships among the images of metric spaces under open boundary-compact maps,pseudo-open boundary-compact maps,open coinpact maps,spaces with point-countable base and some generalized spaces are discussed.The following results are obtained:(1)X has a base which isσ-disjoint at non-isolated points if and only if X is an open,at most boundary-one image of a metric space;(2)Each discretizable space of a space with an uniform base is an open compact and at most boundary-one image of a space with an uniform base.