Topologies generated by discrete subspaces

A topological space X is called discretely generated if for every subset A X we have A = {D : D A and D is a discrete subspace of X}. We say that X is weakly discretely generated if A X and A A implies D A for some discrete D A. It is established that sequential spaces, monotonically normal spaces and compact countably tight spaces are discretely generated. We also prove taht every compact space is weakly discretely generated and under the Continuum Hypothesis any dyadic discretely generated space is metrizable