The $Q$-topology, Whyburn type filters and the cluster set map
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- by Robert A. Herrmann PDF
- Proc. Amer. Math. Soc. 59 (1976), 161-166 Request permission
Erratum: Proc. Amer. Math. Soc. 65 (1977), 375.
Abstract:
We use nonstandard topology and the $Q$-topology to characterize normal, almost-normal, regular, almost-regular, semiregular spaces. The cluster [resp. $\theta$-cluster] set relation is used to characterize regular, almost-regular [resp. strongly-regular] spaces. The Whyburn [resp. Dickman] filter bases are characterized and it is shown that the cluster [resp. $\theta$-cluster] set relation restricted to the domain of the Whyburn [resp. Dickman] filter bases is an essentially continuous [resp. strongly $\theta$-continuous] map iff the space is Hausdorff [resp. Urysohn].References
- Robert Warren Button, Monads for regular and normal spaces, Notre Dame J. Formal Logic 17 (1976), no. 3, 449–456. MR 445483 R. Dickman, $\alpha$-perfect mappings and almost-convergence (preprint). R. Dickman and J. Porter, $\theta$-perfect and $\theta$-absolutely closed functions (preprint).
- R. V. Fuller, A characterization of local compactness, Proc. Amer. Math. Soc. 37 (1973), 615–616. MR 317278, DOI 10.1090/S0002-9939-1973-0317278-9
- Robert A. Herrmann, Nonstandard topological extensions, Bull. Austral. Math. Soc. 13 (1975), no. 2, 269–290. MR 405398, DOI 10.1017/S0004972700024461 —, The $\theta$ and $\alpha$-monads in general topology (to appear).
- W. A. J. Luxemburg, A general theory of monads, Applications of Model Theory to Algebra, Analysis, and Probability (Internat. Sympos., Pasadena, Calif., 1967) Holt, Rinehart and Winston, New York, 1969, pp. 18–86. MR 0244931
- Moshé Machover and Joram Hirschfeld, Lectures on non-standard analysis, Lecture Notes in Mathematics, Vol. 94, Springer-Verlag, Berlin-New York, 1969. MR 0249285, DOI 10.1007/BFb0101447
- Abraham Robinson, Non-standard analysis, North-Holland Publishing Co., Amsterdam, 1966. MR 0205854
- M. K. Singal and Shashi Prabha Arya, On almost-regular spaces, Glasnik Mat. Ser. III 4(24) (1969), 89–99 (English, with Serbo-Croatian summary). MR 243483
- M. K. Singal and Shashi Prabha Arya, Almost normal and almost completely regular spaces, Glasnik Mat. Ser. III 5(25) (1970), 141–152 (English, with Serbo-Croatian summary). MR 275354
- G. T. Whyburn, Directed families of sets and closedness of functions, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 688–692. MR 182953, DOI 10.1073/pnas.54.3.688
- Oswald Wyler, A characterization of regularity in topology, Proc. Amer. Math. Soc. 29 (1971), 588–590. MR 281146, DOI 10.1090/S0002-9939-1971-0281146-X
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 59 (1976), 161-166
- MSC: Primary 54J05
- DOI: https://doi.org/10.1090/S0002-9939-1976-0482730-5
- MathSciNet review: 0482730