Characterizations of ℕ-compactness and realcompactness via ultrafilters in the absence of the axiom of choice
-
Alireza Olfati
and
Eliza Wajch
Abstract
This article concerns the Herrlich-Chew theorem stating that a Hausdorff zero-dimensional space is ℕ-compact if and only if every clopen ultrafilter with the countable intersection property in this space is fixed. It also concerns Hewitt’s theorem stating that a Tychonoff space is realcompact if and only if every z-ultrafilter with the countable intersection property in this space is fixed. The axiom of choice was involved in the original proofs of these theorems. The aim of this article is to show that the Herrlich-Chew theorem is valid in ZF, but it is an open problem if Hewitt’s theorem can be false in a model of ZF. It is proved that Hewitt’s theorem is true in every model of ZF in which the countable axiom of multiple choice is satisfied. A modification of Hewitt’s theorem is given and proved true in ZF. Several applications of the results obtained are shown.
(Communicated by L’ubica Holá)
References
[1] Blair, R. L.: Spaces in which special sets are z-embedded, Canad. J. Math. 28(4) (1976), 673–690.10.4153/CJM-1976-068-9Search in Google Scholar
[2] Chew, K. P.: A characterization of ℕ-compact spaces, Proc. Amer. Math. Soc. 26 (1970), 679–682.10.1090/S0002-9939-1970-0267534-5Search in Google Scholar
[3] Comfort, W. W.: A theorem of Stone-Čech type, and a theorem of Tychonoff type, without the axiom of choice; and their realompact analogues, Fund. Math. 63 (1968), 97–110.10.4064/fm-63-1-97-110Search in Google Scholar
[4] Engelking, R.: General Topology. Sigma Ser. Pure Math. 6, Heldermann, Berlin, 1989.Search in Google Scholar
[5] Engelking, R.—Mrówka, S.: On E-compact spaces, Bull. Acad. Polon. Sci. 6 (1958), 429–436.Search in Google Scholar
[6] Gillman, L.—Jerison, M.: Rings of Continuous Functions, D. Van Nostrand Company, Inc., New York, 1960.10.1007/978-1-4615-7819-2Search in Google Scholar
[7] Good, C.—Tree, I.: Continuing horrors of topology without choice, Topology Appl. 63 (1995), 79–90.10.1016/0166-8641(95)90010-1Search in Google Scholar
[8] Herrlich, H.: S-Compacte Räume, Math. Z. 96 (1967), 228–255.10.1007/BF01124082Search in Google Scholar
[9] Hewitt, E.: Rings of continuous real-valued functions. I, Trans. Amer. Math. Soc. 64 (1948), 45–99.10.1090/S0002-9947-1948-0026239-9Search in Google Scholar
[10] Howard, P.—Rubin, J. E.: Consequences of the Axiom of Choice. Math. Surv. Monogr., Vol. 59, A.M.S., Providence R.I., 1998.10.1090/surv/059Search in Google Scholar
[11] Keremedis, K.—Olfati, A. R.—Wajch, E.: On P-spaces and Gδ-sets in the absence of the Axiom of Choice, Bull. Belg. Math. Soc. Simon Stevin 30 (2023), 194-236.10.36045/j.bbms.230117Search in Google Scholar
[12] Keremedis, K.—Wajch, E.: Hausdorff compactifications in ZF, Topology Appl. 258 (2019), 79–99.10.1016/j.topol.2019.02.046Search in Google Scholar
[13] Keremedis, K.—Wajch, E.: k-Spaces, sequential spaces and related topics in the absence of the axiom of choice, Topology Appl. 318 (2022), Art. ID 108199.10.1016/j.topol.2022.108199Search in Google Scholar
[14] Läuchli, H.: Auswahlaxiom in der Algebra, Comment. Math. Helv. 37 (1962), 1–18.10.1007/BF02566957Search in Google Scholar
[15] Levy, R.—Rice, M. D.: Normal P-spaces and the Gδ-topology, Colloq. Math. 44 (1981), 227–240.10.4064/cm-44-2-227-240Search in Google Scholar
[16] Lorch, E. R.: Compactification, Baire functions and Daniell integration, Acta Sci. Math. (Szeged) 24 (1963), 204–218.Search in Google Scholar
[17] Mrówka, S.: Some properties of Q-spaces, Bull. Acad. Polon. Sci. 5 (1957), 947–950.Search in Google Scholar
[18] Mrówka, S.: On E-compact spaces II, Bull. Acad. Polon. Sci. 14 (1966), 597–605.Search in Google Scholar
[19] Mrówka, S.: Further results on E-compact spaces I, Acta Math. 120 (1968), 161–185.10.1007/BF02394609Search in Google Scholar
[20] Mrówka, S.: Structures of continuous functions I, Acta Math. Acad. Sci. Hung. 21(3–4) (1970), 239–259.10.1007/BF01894771Search in Google Scholar
[21] Mrówka, S.: Recent results on E-compact spaces and structures of continuous functions, Proc. Univ. Oklahoma Topol. Conf. 1972, (1972), 168–221.Search in Google Scholar
[22] Mrówka, S.: Recent results on E-compact spaces. TOPO 72 - General Topology Appl., 2nd Pittsburgh internat. Conf. 1972, Lecture Notes in Math., Vol. 378, 1974, pp. 298–301.10.1007/BFb0068485Search in Google Scholar
[23] Niknejad, J.: Some Properties of Realcompact Spaces and Coarser Normal Spaces, Ph.D. thesis, Department of Mathematics and the Faculty of the Graduate School of the University of Kansas, 2009.Search in Google Scholar
[24] Olfati, A. R.—Wajch, E.: E-compact extensions in the absence of the Axiom of Choice, preprint, https://arxiv.org/abs/2211.00411.Search in Google Scholar
[25] Olfati, A. R.—Wajch, E.: Banaschewski compactifications via special rings of functions in the absence of the axiom of choice, Quaest. Math., to appear.Search in Google Scholar
[26] Porter, J. R.—Woods, G. R.: Extensions and Absolutes of Hausdorff Spaces, Springer, New York, 1988.10.1007/978-1-4612-3712-9Search in Google Scholar
[27] Shirota, T.: A class of topological spaces, Osaka Math. J. 4 (1952), 23–40.Search in Google Scholar
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Articles in the same Issue
- Weak differences, weak BCK-algebras and applications to some partial orders on rings
- Weakly κ-compact topological spaces
- On sharp radius estimates for S*(β) and a product function
- Maximal subextension and stability in m-capacity of maximal subextension of m-subharmonic functions with given boundary values
- Asymptotic behavior of fractional super-linear differential equations
- New and improved oscillation criteria of third-order half-linear delay differential equations via canonical transform
- Global dynamics of the system of difference equations
- Results on oscillatory properties of third-order functional difference equations with semi-canonical operators
- A new approach to metrical fixed point theorems
- Generalized Baker’s result and stability of functional equations using fixed point results
- Characterizations of ℕ-compactness and realcompactness via ultrafilters in the absence of the axiom of choice
- K-theory of oriented flag manifolds
- On certain observations on split continuity and cauchy split continuity
- On the generalized eta- and theta-transformation formulas as the Hecke modular relation
- On some selective star Lindelöf-type properties
