Sequence-covering maps of metric spaces

Let f :X→Y be a map. f is a sequence-covering map if whenever {yn} is a convergent sequence in Y, there is a convergent sequence {xn} in X with each xn∈f−1(yn). f is a 1-sequence-covering map if for each y∈Y, there is x∈f−1(y) such that whenever {yn} is a sequence converging to y in Y there is a sequence {xn} converging to x in X with each xn∈f−1(yn). In this paper we investigate the structure of sequence-covering images of metric spaces, the main results are that (1) sequence-covering, quotient and s-image of a locally separable metric space is a local ℵ0-space; sequence-covering and compact map of a metric space is a 1-sequence-covering map.