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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

When do the upper Kuratowski topology (homeomorphically, Scott topology) and the co-compact topology coincide?


Authors: Szymon Dolecki, Gabriele H. Greco and Alojzy Lechicki
Journal: Trans. Amer. Math. Soc. 347 (1995), 2869-2884
MSC: Primary 54B20; Secondary 06B30, 54A20
DOI: https://doi.org/10.1090/S0002-9947-1995-1303118-7
MathSciNet review: 1303118
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Abstract: A topology is called consonant if the corresponding upper Kuratowski topology on closed sets coincides with the co-compact topology, equivalently if each Scott open set is compactly generated. It is proved that Čechcomplete topologies are consonant and that consonance is not preserved by passage to $ {G_\delta }$-sets, quotient maps and finite products. However, in the class of the regular spaces, the product of a consonant topology and of a locally compact topology is consonant. The latter fact enables us to characterize the topologies generated by some $ \Gamma $-convergences.


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DOI: https://doi.org/10.1090/S0002-9947-1995-1303118-7
Keywords: Kuratowski convergence, co-compact topology, continuous convergence, Gamma convergence, compact families
Article copyright: © Copyright 1995 American Mathematical Society