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 
$S_{\omega}$.
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 $cs$-network 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.
has pullbacks and pushouts.