TOPOLOGIES GENERATED BY DISCRETE SUBSPACES

Abstract

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