Topological group criterion for C ( X ) in compact-open-like topologies, II

We continue from “part I” our address of the following situation. For a Tychonoff space Y, the “second epi-topology” σ is a certain topology on C ( Y ) , which has arisen from the theory of categorical epimorphisms in a category of lattice-ordered groups. The topology σ is always Hausdorff, and σ in...

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Bibliographic Details
Published in:Topology and its applications 2009-10, Vol.156 (16), p.2560-2564
Main Authors: Ball, R., Gochev, V., Hager, A., Zoble, S.
Format: Article
Language:English
Subjects:
Compact-zero topology
Continuum hypothesis
Epi-topology
formula omitted
Polish space
Space with filter
Topological group
Čech–Stone compactification
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Summary:We continue from “part I” our address of the following situation. For a Tychonoff space Y, the “second epi-topology” σ is a certain topology on C ( Y ) , which has arisen from the theory of categorical epimorphisms in a category of lattice-ordered groups. The topology σ is always Hausdorff, and σ interacts with the point-wise addition + on C ( Y ) as: inversion is a homeomorphism and + is separately continuous. When is + jointly continuous, i.e. σ is a group topology? This is so if Y is Lindelöf and Čech-complete, and the converse generally fails. We show in the present paper: under the Continuum Hypothesis, for Y separable metrizable, if σ is a group topology, then Y is (Lindelöf and) Čech-complete, i.e. Polish. The proof consists in showing that if Y is not Čech-complete, then there is a family of compact sets in β Y which is maximal in a certain sense.
ISSN:0166-8641
1879-3207
DOI:10.1016/j.topol.2009.04.008