Abstract:In this note, we introduce the concept of weakly monotonically monolithic spaces, and show that every weakly monotonically monolithic space is a $D$-space. Thus most known conclusions on $D$-spaces can be obtained by this conclusion. As a corollary, we have that if a regular space $X$ is sequential and has a point-countable $wcs^*$-network then $X$ is a $D$-space.
Keywords: $D$-space, sequential space, $wcs^*$-network, weakly monotonically monolithic space
AMS Subject Classification: 54F99 54G99