Journal of Convex Analysis 33 (2026), No. 3&4, 615--628
Copyright Heldermann Verlag 2026
On a Theorem of S.-T. Hu
Gerald Beer
Department of Mathematics, California State University, Los Angeles, U.S.A.
gbeer@cslanet.calstatela.edu
Homeira Pajoohesh
Department of Mathematics, Medgar Evers College, CUNY, Brooklyn, U.S.A.
hpajoohesh@mec.cuny.edu
S.-T. Hu [Boundedness in topological spaces, J. Math. Pures Appl. 228 (1949) 287--320] characterized those bornologies on a metrizable space that are metric bornologies, i.e., that agree with the bornology of metrically bounded subsets as determined by some metric compatible with the topology. We show that the family of such metric bornologies forms a sublattice of the complete lattice of all bornologies on the underlying set which furthermore is stable under denumerable suprema in that larger lattice, but not necessarily denumerable infima. We characterize those metrizable spaces for which the bornology of relatively compact subsets - the intersection of the family of metric bornologies in any case -- is itself a metric bornology. Finally, we characterize those metrizable spaces for which each bornology on the space with a countable closed base (resp. countable open base) is already a metric bornology.
Keywords: Bornology, metric bornology, lattice, lattice of bornologies, Hu's Theorem, relatively compact set, principal bornology.
MSC: 46A17, 06B05, 06B23, 54E45.
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