TOPOLOGIES GENERATED BY DISCRETE SUBSPACES

By A. Dow, Mikhail Tkachenko, V. Tkachuk, R. Wilson. 2002

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A topological space X is called discretely generated if for ev- ery subset A X we have A=(fD:D A and D is a discrete subspace of Xg. We say that X is weakly discretely generated if A X and A6A implies DnA6; for some discrete D A. It is established that sequential spaces, mono- tonically 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 gen- erated space is metrizable.

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