WEAK SEQUENCE-COVERING MAPPING AND CS-NETWORK

A mapping f: X→Y is called weak sequence-covering if whenever {ya} is a sequence in Y converging to y ∈ Y, there exist a subsequence {ynk} and xk∈f^-1(ynk)(k∈N) ,x∈f^-1 (y) such that xk→x. The main results are: (1) Y is a sequential, Frechet, strongly Frechet space iff every weak sepuence-covering mapping onto Y is quotient, pseudo-open, countably bi-quotient respectively, (2) weak sequence-covering mapping preserves cs-network and certain k-(cs-)networks, thus some new mapping theorems on k-(cs-)notworks are proved.

A mapping f: X→Y is called weak sequence-covering if whenever {y_n} is a sequence in Y converging to y∈Y, there exist a subsequence {y_n_k} and x_k∈f^(-1)(y_n_k)(k∈N),x∈f^(-1)(y) such that x_k→x. The main results are: (1) Y is a sequential, Frechet, strongly Frechet space iff every weak sequence-covering mapping onto Y is quotient, pseudo-open, countably bi-quotient respectively, (2) weak sequence-covering mapping preserves cs-network and certain k-(cs-)networks, thus some new mapping theorems on k-(cs-)networks are proved.