On spaces with point-countable $k$-systems

This paper deals with the behavior of M-spaces, countably bi-quasi-k-spaces and singly bi-quasi-k-spaces with point-countable k-systems. For example, we show that every M-space with a point-countable k-system is locally compact paracompact, and every separable singly bi-quasi-k-space with a point-countable k-system has a countable k-system. Also, we consider equivalent relations among spaces entried in Table 1 in Michael's paper (15) when the spaces have point-countable k-systems.