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Keywords:
realcompactness; realcompleteness; uniform space
realcompactness; realcompleteness; uniform space
Summary:
Various characterizations of realcompactness are transferred to uniform spaces giving non-equivalent concepts. Their properties, relations and characterizations are described in this paper. A Shirota-like characterization of certain uniform realcompactness proved by Garrido and Meroño for metrizable spaces is generalized to uniform spaces. The paper may be considered as a unifying survey of known results with some new results added.
References:
[1] Chekeev A.A.: Uniformities for Wallman compactifications and realcompactifications. Topology Appl. 201 (2016), 145–156. DOI 10.1016/j.topol.2015.12.033 | MR 3461161 | Zbl 1342.54021
[2] Chekeev A.A., Kasymova T.J.: Ultrafilter-completeness on zero-sets of uniformly continuous functions. submitted to Proceedings Prague Toposym, 2016.
[3] Engelking R.: General Topology. Heldermann Verlag, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[4] Frolík Z.: A generalization of realcompact spaces. Czechoslovak Math. J. 13 (1963), 127–137. MR 0155289 | Zbl 0112.37603
[5] Garrido M.I., Meroño A.S.: New types of completeness in metric spaces. Ann. Acad. Sci. Fenn. Math. 39 (2014), 733–758. DOI 10.5186/aasfm.2014.3934 | MR 3237048 | Zbl 1303.54010
[6] Garrido M.I., Meroño A.S.: The Samuel realcompactification. Abstracts of Prague Toposym, 2016.
[7] Garrido M.I., Meroño A.S.: The Samuel realcompactification. submitted to Proceedings Prague Toposym, 2016. MR 3688464
[8] Gillman L.: Real-compact spaces (Q-spaces). Bull. Amer. Math. Soc. 63 (1957), 144–145.
[9] Gillman L., Jerison M.: Rings of Continuous Functions. Van Nostrand Co., Princeton, NJ-Toronto-London-New York, 1960. MR 0116199 | Zbl 0327.46040
[10] Hejcman J.: Boundedness in uniform spaces and topological groups. Czechoslovak Math. J. 9 (1959), 544–563. MR 0141075 | Zbl 0132.18202
[11] Hejcman J.: On conservative uniform spaces. Comment. Math. Univ. Carolin. 7 (1966), 411–417. MR 0202107 | Zbl 0181.50902
[12] Herrlich H.: Fortsetzbarkeit stetiger Abbildungenand Kompaktheitsgrad topologischer Räume. Math. Z. 96 (1967), 64–72. DOI 10.1007/BF01111452 | MR 0208560
[13] Hewitt E.: Rings of real-valued continuous functions, I. Trans. Amer. Math. Soc. 64 (1948), 45–99. DOI 10.1090/S0002-9947-1948-0026239-9 | MR 0026239 | Zbl 0032.28603
[14] Hušek M.: The class of k-compact spaces is simple. Math. Z. 110 (1969), 123–126. DOI 10.1007/BF01124977 | MR 0244947 | Zbl 0175.49601
[15] Hušek M., Pulgarín A.: Banach-Stone-like theorems for lattices of uniformly continuous functions. Quaest. Math. 35 (2012), 417–430. DOI 10.2989/16073606.2012.742238 | MR 2999998 | Zbl 1274.54059
[16] Hušek M., Pulgarín A.: When lattices of uniformly continuous functions on $X$ determine $X$. Topology Appl. 194 (2015), 228–240. MR 3404615 | Zbl 1328.54013
[17] Isbell J.R.: Euclidean and weak uniformities. Pacific J. Math. 8 (1958), 67–86. DOI 10.2140/pjm.1958.8.67 | MR 0097794 | Zbl 0081.16802
[18] Isbell J.R.: Algebras of uniformly continuous functions. Ann. of Math. 68 (1958), 96–125. DOI 10.2307/1970045 | MR 0103407 | Zbl 0081.11101
[19] Isbell J.R.: Uniform Spaces. Math. Surveys, 12, American Mathematical Society, Rhode Island, 1964. MR 0170323 | Zbl 0124.15601
[20] Katětov M.: On real-valued functions on topological spaces. Fund. Math. 38 (1951), 85–91. DOI 10.4064/fm-38-1-85-91 | MR 0050264
[21] Mrówka S.: Some properties of Q-spaces. Bull. Acad. Polon. Sci. Ser. Math. 5 (1957), 947–950. MR 0095465 | Zbl 0079.38602
[22] Mrówka S.: An elementary proof of Katětov's theorem concerning Q-spaces. Michigan Math. J. 11 (1964), 61–63. DOI 10.1307/mmj/1028999035 | MR 0161308 | Zbl 0117.16002
[23] Nachbin L.: Topological vector spaces of continuous functions. Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 471–474. DOI 10.1073/pnas.40.6.471 | MR 0063647 | Zbl 0055.09803
[24] Njastad O.: On real-valued proximity mappings. Math. Ann. 154 (1964), 413–419. DOI 10.1007/BF01375524 | MR 0166757 | Zbl 0121.39401
[25] Pelant J.: Reflections not preserving completeness. Seminar Uniform Spaces (Prague 1973–1974), pp. 235–240. MR 0474219 | Zbl 0327.54024
[26] Rice M.D., Reynolds G.D.: Completeness and covering properties of uniform spaces. Quart. J. Math. Oxford 29 (1978), 367–374. DOI 10.1093/qmath/29.3.367 | MR 0509702 | Zbl 0411.54028
[27] Rice M.D.: Subcategories of uniform spaces. Trans. Amer. Math. Soc. 201 (1975), 305–314. DOI 10.1090/S0002-9947-1975-0358708-2 | MR 0358708 | Zbl 0615.54019
[28] Shirota T.: On spaces with a complete structure. Proc. Japan Acad. 27 (1951), 513–516. MR 0048781 | Zbl 0045.25702
[29] Shirota T.: A class of topological spaces. Osaka Math. J. 4 (1952), 23–40. MR 0050872 | Zbl 0047.41704
[30] Weir M.D.: Hewitt-Nachbin Spaces. North Holland, Amsterdam, 1975. MR 0514909 | Zbl 0314.54002
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