Ultrafilters: Some old and some new results
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- by W. W. Comfort PDF
- Bull. Amer. Math. Soc. 83 (1977), 417-455
References
- A. V. Arhangel′skiĭ, An extremal disconnected bicompactum of weight ${\mathfrak {c}}$ is inhomogeneous, Dokl. Akad. Nauk SSSR 175 (1967), 751–754 (Russian). MR 0219039
- Hewitt Kenyon and I. N. Baker, Advanced Problems and Solutions: Solutions: 5077, Amer. Math. Monthly 71 (1964), no. 2, 219–220. MR 1532554, DOI 10.2307/2311775
- James E. Baumgartner, A short proof of Hindman’s theorem, J. Combinatorial Theory Ser. A 17 (1974), 384–386. MR 354394, DOI 10.1016/0097-3165(74)90103-4 Be Murray Bell, manuscript.
- Allen R. Bernstein, A new kind of compactness for topological spaces, Fund. Math. 66 (1969/70), 185–193. MR 251697, DOI 10.4064/fm-66-2-185-193 Bl Andreas Raphael Blass, Orderings of ultrafilters, Doctoral Dissertation, Harvard Univ., 1970. Bo David D. Booth, Countably indexed ultrafilters, Doctoral Dissertation, Univ. of Wisconsin, 1969.
- N. G. de Bruijn and P. Erdös, A colour problem for infinite graphs and a problem in the theory of relations, Nederl. Akad. Wetensch. Proc. Ser. A. 54 = Indagationes Math. 13 (1951), 369–373. MR 0046630
- Eduard Čech, On bicompact spaces, Ann. of Math. (2) 38 (1937), no. 4, 823–844. MR 1503374, DOI 10.2307/1968839
- Paul J. Cohen, Set theory and the continuum hypothesis, W. A. Benjamin, Inc., New York-Amsterdam, 1966. MR 0232676
- W. W. Comfort, A theorem of Stone-Čech type, and a theorem of Tychonoff type, without the axiom of choice; and their realcompact analogues, Fund. Math. 63 (1968), 97–110. MR 236880, DOI 10.4064/fm-63-1-97-110 C2 W. W. Comfort, Review of [GS]. MR 52 # 1633.
- W. W. Comfort, Some recent applications of ultrafilters to topology, General topology and its relations to modern analysis and algebra, IV (Proc. Fourth Prague Topological Sympos., Prague, 1976) Lecture Notes in Math., Vol. 609, Springer, Berlin, 1977, pp. 34–42. MR 0451187
- W. W. Comfort and S. Negrepontis, On families of large oscillation, Fund. Math. 75 (1972), no. 3, 275–290. MR 305343, DOI 10.4064/fm-75-3-275-290
- W. W. Comfort and S. Negrepontis, The theory of ultrafilters, Die Grundlehren der mathematischen Wissenschaften, Band 211, Springer-Verlag, New York-Heidelberg, 1974. MR 0396267, DOI 10.1007/978-3-642-65780-1
- W. W. Comfort and S. Negrepontis, Continuous pseudometrics, Lecture Notes in Pure and Applied Mathematics, Vol. 14, Marcel Dekker, Inc., New York, 1975. MR 0410618
- J. N. Crossley, C. J. Ash, C. J. Brickhill, J. C. Stillwell, and N. H. Williams, What is mathematical logic?, Oxford Paperbacks University Series, No. 60, Oxford University Press, London-New York, 1972. MR 0414308 vD1 Eric K. van Douwen, Functions from the integers to the integers and topology (manuscript).
- Eric K. van Douwen, Why certain Čech-Stone remainders are not homogeneous, Colloq. Math. 41 (1979), no. 1, 45–52. MR 550626, DOI 10.4064/cm-41-1-45-52 Ef1 B. Efimov, Extremally disconnected bicompacta of continuum π-weight, Dokl. Akad. Nauk SSSR 183 (1968), 511-514 = Soviet Math. Dokl. 9 (1968), 1404-1407. MR 38 #5172.
- B. A. Efimov, On the imbedding of extremally disconnected spaces into bicompacta, General topology and its relations to modern analysis and algebra, III (Proc. Third Prague Topological Sympos., 1971) Academia, Prague, 1972, pp. 103–107. MR 0355966
- Robert Ellis, Lectures on topological dynamics, W. A. Benjamin, Inc., New York, 1969. MR 0267561
- Ryszard Engelking, Zarys topologii ogólnej, Biblioteka Matematyczna, Tom 25, Państwowe Wydawnictwo Naukowe, Warsaw, 1965 (Polish). MR 0221456 FK G. Fichtenholz and L. Kantorovitch, Sur les opérations linéaires dans l’espace des fonctions bornées, Studia Math. 5 (1935), 69-98. F1 Zdenĕk Frolík, Types of ultrafilters on countable sets, General Topology and its Relations to Modem Analysis and Algebra, II (Proc. Second Prague Topological Sympos., 1966), Academia, Prague, 1967. MR 38 #654.
- Zdeněk Frolík, Sums of ultrafilters, Bull. Amer. Math. Soc. 73 (1967), 87–91. MR 203676, DOI 10.1090/S0002-9904-1967-11653-7
- Zdeněk Frolík, Non-homogeneity of $\beta P-P$, Comment. Math. Univ. Carolinae 8 (1967), 705–709. MR 266160
- Zdeněk Frolík, Homogeneity problems for extremally disconnected spaces, Comment. Math. Univ. Carolinae 8 (1967), 757–763. MR 264584
- Z. Frolík, Maps of extremally disconnected spaces, theory of types, and applications, General Topology and its Relations to Modern Analysis and Algebra, III (Proc. Conf., Kanpur, 1968) Academia, Prague, 1971, pp. 131–142. MR 0295305
- Leonard Gillman and Melvin Henriksen, Rings of continuous functions in which every finitely generated ideal is principal, Trans. Amer. Math. Soc. 82 (1956), 366–391. MR 78980, DOI 10.1090/S0002-9947-1956-0078980-4
- 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
- John Ginsburg and Victor Saks, Some applications of ultrafilters in topology, Pacific J. Math. 57 (1975), no. 2, 403–418. MR 380736, DOI 10.2140/pjm.1975.57.403
- R. L. Graham and B. L. Rothschild, Ramsey’s theorem for $n$-parameter sets, Trans. Amer. Math. Soc. 159 (1971), 257–292. MR 284352, DOI 10.1090/S0002-9947-1971-0284352-8
- A. Hajnal, Proof of a conjecture of S. Ruziewicz, Fund. Math. 50 (1961/62), 123–128. MR 131986, DOI 10.4064/fm-50-2-123-128 Hs Felix Hausdorff, Über zwei Sätze von G. Fichtenholz und L. Kantorovitch, Studia Math. 6 (1936), 18-19.
- Edwin Hewitt, A remark on density characters, Bull. Amer. Math. Soc. 52 (1946), 641–643. MR 17329, DOI 10.1090/S0002-9904-1946-08613-9
- Neil Hindman, The existence of certain ultra-filters on $N$ and a conjecture of Graham and Rothschild, Proc. Amer. Math. Soc. 36 (1972), 341–346. MR 307926, DOI 10.1090/S0002-9939-1972-0307926-0
- Neil Hindman, Finite sums from sequences within cells of a partition of $N$, J. Combinatorial Theory Ser. A 17 (1974), 1–11. MR 349574, DOI 10.1016/0097-3165(74)90023-5
- Takesi Isiwata, A generalization of Rudin’s theorem for the homogeneity problem, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 5 (1957), 300–303. MR 95463
- Thomas J. Jech, The axiom of choice, Studies in Logic and the Foundations of Mathematics, Vol. 75, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. MR 0396271
- I. Juhas, Remarks on a theroem of B. Pospíšil, Comment. Math. Univ. Carolinae 8 (1967), 231–247 (Russian). MR 216467 J2 I. Juhász, On the character of points in βNm, Contributions to Extension Theory of Topological Structures (Proc. Berlin Sympos. 1967), VEB Deutscher Verlag Wiss., Berlin, 1969, pp. 139-140. JNW I. Juhász, Zs. Nagy and William Weiss, On countably compact, locally compact spaces. Math. Institute Hungarian Acad. Sci., 1976, 26 pp. (preprint).
- M. Katětov, Characters and types of point sets, Fund. Math. 50 (1961/62), 369–380 (Russian). MR 137094
- Miroslav Katětov, Products of filters, Comment. Math. Univ. Carolinae 9 (1968), 173–189. MR 250257
- M. Katětov, On descriptive classification of functions, General topology and its relations to modern analysis and algebra, III (Proc. Third Prague Topological Sympos., 1971) Academia, Prague, 1972, pp. 235–242. MR 0355956 K H. Jerome Keisler, mimeographed lecture notes, Univ. of California, Los Angeles, 1967.
- H. J. Keisler and A. Tarski, From accessible to inaccessible cardinals. Results holding for all accessible cardinal numbers and the problem of their extension to inaccessible ones, Fund. Math. 53 (1963/64), 225–308. MR 166107, DOI 10.4064/fm-53-3-225-308
- John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144 Kn Hewitt Kenyon, Problem 5077, Amer. Math. Monthly 70 (1963), 216. Ku1 Kenneth Kunen, On the compactification of the integers, Notices Amer. Math. Soc. 17 (1970), 299. Abstract #70T-G7.
- Kenneth Kunen, Ultrafilters and independent sets, Trans. Amer. Math. Soc. 172 (1972), 299–306. MR 314619, DOI 10.1090/S0002-9947-1972-0314619-7
- Kenneth Kunen, Some points in $\beta N$, Math. Proc. Cambridge Philos. Soc. 80 (1976), no. 3, 385–398. MR 427070, DOI 10.1017/S0305004100053032 Ku4 Hewitt Kenyon, manuscript.
- Edward Marczewski, Séparabilité et multiplication cartésienne des espaces topologiques, Fund. Math. 34 (1947), 127–143 (French). MR 21680, DOI 10.4064/fm-34-1-127-143
- S. Mrówka, On the potency of subsets of $\beta N$, Colloq. Math. 7 (1959), 23–25. MR 114193, DOI 10.4064/cm-7-1-23-25 N1 Stelios A. Negrepontis, A homology theory for realcompact spaces, Doctoral Dissertation, Univ. of Rochester, 1965.
- S. Negrepontis, The existence of certain uniform ultrafilters, Ann. of Math. (2) 90 (1969), 23–32. MR 246777, DOI 10.2307/1970679 O John V. William O’Callaghan, Endohomeomorphisms and compactness properties of products and generalized Σ-products, Doctoral Dissertation, Wesleyan Univ., 1976.
- E. S. Pondiczery, Power problems in abstract spaces, Duke Math. J. 11 (1944), 835–837. MR 11104, DOI 10.1215/S0012-7094-44-01171-3 P1 Bedřich Pospíšil, Remark on bicompact spaces, Ann. of Math. (2) 38 (1937), 485-486.
- Bedřich Pospíšil, On bicompact spaces, Publ. Fac. Sci. Univ. Masaryk 1939 (1939), no. 270, 16. MR 1454
- M. Rajagopalan, Some outstanding problems in topology and the $V$-process, Categorical topology (Proc. Conf., Mannheim, 1975) Lecture Notes in Math., Vol. 540, Springer, Berlin, 1976, pp. 501–517. MR 0448323
- M. Rajagopalan and R. Grant Woods, Products of sequentially compact spaces and the $V$-process, Trans. Amer. Math. Soc. 232 (1977), 245–253. MR 451219, DOI 10.1090/S0002-9947-1977-0451219-7 Ra F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. (2) 30 (1930), 264-286. Ri F. Riesz, Stetigkeitsbegriff und abstrakte Mengenlehre, Atti IV Congresso Internaz. Mat., Vol. II (Roma, 1908), R. Accad. Lincei, Rome, 1909, pp. 18-24. Ro Frederick Rowbottom, Some strong axioms of infinity incompatible with the axiom of constructibility, Doctoral Dissertation, Univ. of Wisconsin, 1964.
- Mary Ellen Rudin, Types of ultrafilters, Topology Seminar (Wisconsin, 1965) Ann. of Math. Studies, No. 60, Princeton Univ. Press, Princeton, N.J., 1966, pp. 147–151. MR 0216451
- Mary Ellen Rudin, Partial orders on the types in $\beta N$, Trans. Amer. Math. Soc. 155 (1971), 353–362. MR 273581, DOI 10.1090/S0002-9947-1971-0273581-5
- Walter Rudin, Homogeneity problems in the theory of Čech compactifications, Duke Math. J. 23 (1956), 409–419. MR 80902 Sa1 Victor Saks, Countably compact groups, Doctoral Dissertation, Wesleyan Univ., 1972.
- Victor Saks, Ultrafilter invariants in topological spaces, Trans. Amer. Math. Soc. 241 (1978), 79–97. MR 492291, DOI 10.1090/S0002-9947-1978-0492291-9
- Victor Saks and R. M. Stephenson Jr., Products of ${\mathfrak {m}}$-compact spaces, Proc. Amer. Math. Soc. 28 (1971), 279–288. MR 273570, DOI 10.1090/S0002-9939-1971-0273570-6
- Sergio Salbany, On compact$^{\ast }$ spaces and compactifications, Proc. Amer. Math. Soc. 45 (1974), 274–280. MR 355970, DOI 10.1090/S0002-9939-1974-0355970-1
- C. T. Scarborough and A. H. Stone, Products of nearly compact spaces, Trans. Amer. Math. Soc. 124 (1966), 131–147. MR 203679, DOI 10.1090/S0002-9947-1966-0203679-7
- Z. Semadeni, Periods of measurable functions and the Stone-Čech compactification, Amer. Math. Monthly 71 (1964), 891–893. MR 170992, DOI 10.2307/2312401 Si Waclaw Sierpiński, Fonctions additives non complètement additives et fonctions non mesurables, Fund. Math. 30 (1938), 96-99.
- Robert M. Solovay, A model of set-theory in which every set of reals is Lebesgue measurable, Ann. of Math. (2) 92 (1970), 1–56. MR 265151, DOI 10.2307/1970696
- M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), no. 3, 375–481. MR 1501905, DOI 10.1090/S0002-9947-1937-1501905-7 U1 Stanislaw Ulam, Concerning functions of sets, Fund. Math. 14 (1929), 231-233. U2 Stanislaw Ulam, Zur Masstheorie in der allgemeinen Mengenlehre, Fund. Math. 16 (1930), 140-150. V1 J. E. Vaughan, Some examples concerning α-bounded spaces, Proc. Ohio Univ., Institute for Medicine and Mathematics, Topology Conf. (to appear).
- J. E. Vaughan, Products of perfectly normal, sequentially compact spaces, J. London Math. Soc. (2) 14 (1976), no. 3, 517–520. MR 464163, DOI 10.1112/jlms/s2-14.3.517 W1 A. D. Wallace, A note on mobs. I, II, An. Acad. Brasil. Ci. 24 (1952), 329-334; ibid. 25 (1953), 335-336. MR 14, 724; 15, 854.
- A. D. Wallace, The structure of topological semigroups, Bull. Amer. Math. Soc. 61 (1955), 95–112. MR 67907, DOI 10.1090/S0002-9904-1955-09895-1
Additional Information
- Journal: Bull. Amer. Math. Soc. 83 (1977), 417-455
- MSC (1970): Primary 04A20, 04A30, 10A45, 54A25, 54C45, 54D20, 54G05
- DOI: https://doi.org/10.1090/S0002-9904-1977-14316-4
- MathSciNet review: 0454893