SEQUENCE-COVERING MAPS ON GENERALIZED METRIC SPACES

摘要

Let f: X → Y be a map. f is a sequence-covering map [25] if whenever {y_n} is a convergent sequence in Y there is a convergent sequence {x_n} in X with each x_n ∈ f~(-1)(y_n)； f is an 1-sequence-covering map [14] if for each y ∈ Y there is x ∈ f~(-1)(y) such that whenever {y_n} is a sequence converging to y in Y there is a sequence {xn} converging to x in X with each x_n ∈ f~(-1)(y_n). In this paper， we mainly discuss the sequence-covering maps on generalized metric spaces, and give an affirmative answer to a question in [13] and some related questions, which improve some results in [13, 16, 28], respectively. Moreover, we also prove that open and closed maps preserve strongly monotonically monolithity, and closed sequence-covering maps preserve spaces with a σ-point-discrete κ-network. Some questions about sequence-covering maps on generalized metric spaces are posed.