Abstract
In this paper, we give counterexamples of some questions on generalized
metric spaces. First we show that there exists an open sequence-covering map of a
countable g-second countable space onto the sequential fan Sω. This is a counterexample
for a question posed by Y. Tanaka. Second we show that there exists a regular
Fr´echet space Y satisfying the following conditions: (1) Y has a point-countable csnetwork
and k-network of closed subsets; (2) every first countable closed subset of
Y is countable; (3) Y is not locally separable and does not have any star-countable
k-network. This is a counterexample for questions posed by S. Lin.
