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Topologies generated by discrete subspaces. (English) Zbl 1009.54005
A topological space $$X$$ is called discretely generated if for every $$A\subset X$$ the closure of $$A$$ is the union of the closures of discrete subspaces of $$A$$, and $$X$$ is a weakly discretely generated space if for every $$A\subset X$$ with $$\overline{A}\neq A$$ there exists a discrete subset $$D$$ of $$A$$ such that $$\overline{D}\smallsetminus A\neq\emptyset$$. The authors prove that sequential spaces, monotonically normal spaces and compact spaces with countable tightness are discretely generated. Also, such is a regular space $$X$$ which has a nested local base at any point of $$X$$. The authors establish that under the Continuum Hypothesis any discretely generated dyadic compact space is metrizable. They show that there exist pseudocompact Tikhonov spaces in ZFC which are not weakly discretely generated. The authors prove that any Tikhonov countably compact space of weight $$\leq\omega_1$$ is weakly discretely generated while there are models of ZFC with countably compact Tikhonov spaces of weight $$\omega_2$$ which fail to be weakly discretely generated. The paper closes with seven open problems, which indicate a natural line of further investigations on discretely and weakly discretely generated spaces.

MSC:
 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets) 54A35 Consistency and independence results in general topology 54D30 Compactness 54D55 Sequential spaces 54E35 Metric spaces, metrizability