On I-covering images of metric spaces

Let I be an ideal on N. A mapping f : X ? Y is called an I-covering mapping provided a sequence {yn}n?N is I-converging to a point y in Y, there is a sequence {xn}n?N converging to a point x in X such that x ? f ?1(y) and each xn ? f ?1(yn). In this paper we study the spaces with certain I-cs-networks and investigate the characterization of the images of metric spaces under certain I-covering mappings, which prompts us to discover I-cs f -networks. The following main results are obtained: (1) A space X has an I-cs f -network if and only if X is a continuous and I-covering image of a metric space. (2) A space X is an I-cs f -countable space if and only if X is a continuous I-covering and boundary s-image of a metric space. (3) A space X has a point-countable I-cs-network if and only if X is a continuous I-covering and s-image of a metric space.