## The Mackey problem for the compact-open topology

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- by Robert F. Wheeler PDF
- Trans. Amer. Math. Soc.
**222**(1976), 255-265 Request permission

## Abstract:

Let ${C_c}(T)$ denote the space of continuous real-valued functions on a completely regular Hausdorff space*T*, endowed with the compact-open topology. Well-known results of Nachbin, Shirota, and Warner characterize those

*T*for which ${C_c}(T)$ is bornological, barrelled, and infrabarrelled. In this paper the question of when ${C_c}(T)$ is a Mackey space is examined. A convex strong Mackey property (CSMP), intermediate between infrabarrelled and Mackey, is introduced, and for several classes of spaces (including first countable and scattered spaces), a necessary and sufficient condition on

*T*for ${C_c}(T)$ to have CSMP is obtained.

## References

- Abdel Ghayoum A. Babiker,
*On almost discrete spaces*, Mathematika**18**(1971), 163–167. MR**296238**, DOI 10.1112/S0025579300005404 - Philip Bacon,
*The compactness of countably compact spaces*, Pacific J. Math.**32**(1970), 587–592. MR**257975**, DOI 10.2140/pjm.1970.32.587 - Henri Buchwalter,
*Fonctions continues et mesures sur un espace complètement régulier*, Summer School on Topological Vector Spaces (Univ. Libre Bruxelles, Brussels, 1972) Lecture Notes in Math., Vol. 331, Springer, Berlin, 1973, pp. 183–202 (French). MR**0482104** - R. Creighton Buck,
*Bounded continuous functions on a locally compact space*, Michigan Math. J.**5**(1958), 95–104. MR**105611** - John B. Conway,
*The strict topology and compactness in the space of measures. II*, Trans. Amer. Math. Soc.**126**(1967), 474–486. MR**206685**, DOI 10.1090/S0002-9947-1967-0206685-2 - M. De Wilde and J. Schmets,
*Caractérisation des espaces $C(X)$ ultrabornologiques*, Bull. Soc. Roy. Sci. Liège**40**(1971), 119–121 (French, with English summary). MR**291781** - James Dugundji,
*Topology*, Allyn and Bacon, Inc., Boston, Mass., 1966. MR**0193606** - D. H. Fremlin, D. J. H. Garling, and R. G. Haydon,
*Bounded measures on topological spaces*, Proc. London Math. Soc. (3)**25**(1972), 115–136. MR**344405**, DOI 10.1112/plms/s3-25.1.115 - Leonard Gillman and Meyer Jerison,
*Rings of continuous functions*, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR**0116199**, DOI 10.1007/978-1-4615-7819-2 - S. L. Gulden, W. M. Fleischman, and J. H. Weston,
*Linearly ordered topological spaces*, Proc. Amer. Math. Soc.**24**(1970), 197–203. MR**250272**, DOI 10.1090/S0002-9939-1970-0250272-2
D. Gulick, - Denny Gulick,
*Duality theory for the strict topology*, Studia Math.**49**(1973/74), 195–208. MR**333688**, DOI 10.4064/sm-49-3-195-208 - Denny Gulick,
*The $\sigma$-compact-open topology and its relatives*, Math. Scand.**30**(1972), 159–176. MR**331031**, DOI 10.7146/math.scand.a-11072 - Richard Haydon,
*On compactness in spaces of measures and measurecompact spaces*, Proc. London Math. Soc. (3)**29**(1974), 1–16. MR**361745**, DOI 10.1112/plms/s3-29.1.1 - Richard Haydon,
*Sur un problème de H. Buchwalter*, C. R. Acad. Sci. Paris Sér. A-B**275**(1972), A1077–A1080 (French). MR**315382** - Richard Haydon,
*Trois exemples dans la théorie des espaces de fonction continues*, C. R. Acad. Sci. Paris Sér. A-B**276**(1973), A685–A687 (French). MR**326331** - Takesi Isiwata,
*On closed countably-compactifications*, General Topology and Appl.**4**(1974), 143–167. MR**345072**, DOI 10.1016/0016-660X(74)90017-8 - J. L. Kelley and Isaac Namioka,
*Linear topological spaces*, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J., 1963. With the collaboration of W. F. Donoghue, Jr., Kenneth R. Lucas, B. J. Pettis, Ebbe Thue Poulsen, G. Baley Price, Wendy Robertson, W. R. Scott, Kennan T. Smith. MR**0166578**, DOI 10.1007/978-3-662-41914-4 - J. D. Knowles,
*On the existence of non-atomic measures*, Mathematika**14**(1967), 62–67. MR**214719**, DOI 10.1112/S0025579300008020 - Peter D. Morris and Daniel E. Wulbert,
*Functional representation of topological algebras*, Pacific J. Math.**22**(1967), 323–337. MR**213876**, DOI 10.2140/pjm.1967.22.323 - Steven E. Mosiman and Robert F. Wheeler,
*The strict topology in a completely regular setting: relations to topological measure theory*, Canadian J. Math.**24**(1972), 873–890. MR**328567**, DOI 10.4153/CJM-1972-087-2 - Leopoldo Nachbin,
*Topological vector spaces of continuous functions*, Proc. Nat. Acad. Sci. U.S.A.**40**(1954), 471–474. MR**63647**, DOI 10.1073/pnas.40.6.471 - M. Rajagopalan and R. F. Wheeler,
*Sequential compactness of $X$ implies a completeness property for $C(X)$*, Canadian J. Math.**28**(1976), no. 1, 207–210. MR**405341**, DOI 10.4153/CJM-1976-026-9 - Mary Ellen Rudin,
*Lectures on set theoretic topology*, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 23, Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, Providence, R.I., 1975. Expository lectures from the CBMS Regional Conference held at the University of Wyoming, Laramie, Wyo., August 12–16, 1974. MR**0367886**, DOI 10.1090/cbms/023 - Walter Rudin,
*Continuous functions on compact spaces without perfect subsets*, Proc. Amer. Math. Soc.**8**(1957), 39–42. MR**85475**, DOI 10.1090/S0002-9939-1957-0085475-7 - Helmut H. Schaefer,
*Topological vector spaces*, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1966. MR**0193469** - F. Dennis Sentilles,
*Bounded continuous functions on a completely regular space*, Trans. Amer. Math. Soc.**168**(1972), 311–336. MR**295065**, DOI 10.1090/S0002-9947-1972-0295065-1 - Taira Shirota,
*On locally convex vector spaces of continuous functions*, Proc. Japan Acad.**30**(1954), 294–298. MR**64389**
F. D. Tall, - V. S. Varadarajan,
*Measures on topological spaces*, Mat. Sb. (N.S.)**55 (97)**(1961), 35–100 (Russian). MR**0148838** - Seth Warner,
*The topology of compact convergence on continuous function spaces*, Duke Math. J.**25**(1958), 265–282. MR**102735**
W. A. R. Weiss, - Robert F. Wheeler,
*Well-behaved and totally bounded approximate identities for $C_{0}(X)$*, Pacific J. Math.**65**(1976), no. 1, 261–269. MR**458150**, DOI 10.2140/pjm.1976.65.261

*Duality theory for spaces of continuous functions*(to appear).

*An alternative to the continuum hypothesis and its uses in general topology*(to appear).

*Countably compact, perfectly normal spaces may or may not be compact*, Notices Amer. Math. Soc.

**22**(1975), A-334. Abstract # 75T-G32.

## Additional Information

- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**222**(1976), 255-265 - MSC: Primary 46E10
- DOI: https://doi.org/10.1090/S0002-9947-1976-0425593-0
- MathSciNet review: 0425593