The dual of (Cb (X), tσ ): measures and Gulick’s conjecture

Authors

  • Subiman Kundu Former Professor, Department of Mathematics Indian Institute of Technology Delhi, New Delhi 110016, India
  • Manisha Aggarwal Assistant Professor, Department of Mathematical and Computational Sciences National Institute of Technology Karnataka, Surathkal, Mangaluru 575025, India
  • Alexander V. Osipov Head Scientist Researcher, Krasovskii Institute of Mathematics and Mechanics Professor, Ural Federal University, 620990, Yekaterinburg, Russia

DOI:

https://doi.org/10.17398/

Keywords:

Function space, σ-compact-open topology, Dual, measures, Borel sets

Abstract

In [4], D. Gulick introduced a new topology, σ-compact-open topology tσ on Cb(X), where Cb(X) is the set of all real-valued bounded continuous functions on a Tychonoff space X. This space was denoted (Cb(X), tσ ) by him and he studied its dual space. First we give an elegant measure-theoretic characterization of this dual space. Gulick compared this dual space with the dual of (Cb(X), tn ), where tn is the topology on Cb(X) generated by the supremum norm ∥ · ∥∞  on Cb(X). Since tσ ⊆ tn , the dual (Cb(X), tσ )* ⊆ (Cb(X), tn )* . Gulick has conjectured that for a locally compact Hausdorff space X, if the duals are same, then tσ = tn . By constructing a complicated locally compact Hausdorff space X, we show that the duals are the same but tσ is strictly weaker than tn.

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References

[1] C.D. Aliprantis, O. Burkinshaw, “Principles of real analysis”, Third edition. Academic Press, Inc., San Diego, CA, 1998.

[2] R. Engelking, “General topology”, Translated from the Polish by the author. Second edition. Sigma Ser. Pure Math., 6. Heldermann Verlag, Berlin, 1989.

[3] L. Gillman, M. Jerison, “Rings of continuous functions”, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960.

[4] D. Gulick, The σ-compact-open topology and its relatives, Math. Scand. 30 (1972), 159 – 176.

[5] S. Kundu, A. Okuyama, Complete duals of C ∗ (X), Math. Scand. 72 (1) (1993), 33 – 46.

[6] S. Kundu, R.A. McCoy, Topologies between compact and uniform convergence on function spaces, Internat. J. Math. Math. Sci. 16 (1) (1993), 101 – 110.

[7] S. Kundu, V. Pandey, The metrizability and completeness of the σ-compact-open topology on C ∗ (X), Topology Appl. 159 (3) (2012), 593 – 602.

[8] L.A. Steen, J.A. Seebach, Jr., “Counterexamples in topology”, Second edition, Springer-Verlag, New York-Heidelberg, 1978.

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Published

2026-11-10

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How to Cite

The dual of (Cb (X), tσ ): measures and Gulick’s conjecture. (2026). Extracta Mathematicae, 9 pp. . https://doi.org/10.17398/