Ultrafilters: Some old and some new results

BY W. W. COMFORT I am grateful to Peter Freyd for his generous introductory comments, and I am grateful also to the [Selection] Committee for extending the invitation to speak to you. Last month a colleague with whom I attempted to discuss today's remarks suggested drily that as half of the team responsible for [CN2] I have probably already said more than enough about ultrafllters, and that if I insist on pursuing the matter further today I could do so most gracefully and efficiently simply by offering a complimentary copy of [CN2] to each of you. Eschewing that advice I shall in the hour allotted to me attempt to achieve the following three goals. (A) To acquaint you with what I think are some of the most basic, fundamental facts about ultrafllters on a discrete topological space; this material is sufficiently simple and elegant that it can be absorbed comfortably into a first-year graduate course in general topology. (B) To give some partial results, less definitive and less conclusive than the optimal theorems available, concerning the existence of particular ultrafllters with special properties; I hope that the results chosen in this connection have the complementary virtues that they are sufficiently powerful to handle most of the situations treated by the more powerful results which we shall ignore, and that their proofs are significantly simpler than those of the more general results. (C) To record some results about ultrafllters which came to my attention after the publication of [CN2]; I have chosen today to emphasize three relatively new results which are not formally concerned with ultrafllters and which indeed make no mention of ultrafllters in their statements, but which nevertheless have been given proofs in which ultrafllters play an important catalytic role. My hope is that (even) those of you not professionally inclined toward topology or set theory will find something potentially useful, or amusing, among the basic results given in (A). The theorems selected for inclusion in (B) are given not only because of the intrinsic beauty and elegance of their proofs, but also because they serve to indicate the principal sorts of questions

[1]  Z. Semadeni.  Periods of Measurable Functions and the Stone-Cech Compactification , 1964 .

[2]  A. Bernstein.  A new kind of compactness for topological spaces , 1970 .

[3]  Ryszard Engelking,et al.  Outline of general topology , 1968 .

[4]  Frank Plumpton Ramsey,et al.  On a Problem of Formal Logic , 1930 .

[5]  R. Graham,et al.  Ramsey’s theorem for $n$-parameter sets , 1971 .

[6]  Wacław Sierpiński,et al.  Fonctions additives non complètement additives et fonctions non mesurables , 1938 .

[7]  Alfred Tarski,et al.  From accessible to inaccessible cardinals (Results holding for all accessible cardinal numbers and the problem of their extension to inaccessible ones) , 1964 .

[8]  R. Solovay.  A model of set-theory in which every set of reals is Lebesgue measurable* , 1970 .

[9]  James E. Baumgartner,et al.  A Short Proof of Hindman's Theorem , 1974, J. Comb. Theory, Ser. A.

[10]  A. Hajnal,et al.  Proof of a conjecture of B. Ruziewicz , 1961 .

[11]  J. Ginsburg,et al.  Some applications of ultrafilters in topology , 1975 .

[12]  TYPES OF ULTRAFILTERS , 1967 .

[13]  E. Marczewski.  Séparabilité et multiplication cartésienne des espaces topologiques , 1947 .

[14]  P. J. Cohen.  Set Theory and the Continuum Hypothesis , 1966 .

[15]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[16]  M. Rajagopalan.  Some outstanding problems in topology and the V-process , 1976 .

[17]  Sergio Salbany,et al.  On compact* spaces and compactifications , 1974 .

[18]  V. Saks.  Ultrafilter invariants in topological spaces , 1978 .

[19]  A. H. Stone,et al.  Products of nearly compact spaces , 1966 .

[20]  E. Čech.  On Bicompact Spaces , 1937 .

[21]  A. D. Wallace.  The structure of topological semigroups , 1955 .

[22]  V. Saks,et al.  Products of M-Compact Spaces , 1971 .

[23]  Leonard Gillman,et al.  Rings of continuous functions , 1961 .

[24]  E. S. Pondiczery.  Power problems in abstract spaces , 1944 .

[25]  Z. Frolík.  Homogeneity problems for extremally disconnected spaces , 1967 .

[26]  Neil Hindman,et al.  Finite Sums from Sequences Within Cells of a Partition of N , 1974, J. Comb. Theory, Ser. A.

[27]  Zdeněk Frolík,et al.  Sums of ultrafilters , 1967 .

[28]  M. Henriksen,et al.  Rings of continuous functions in which every finitely generated ideal is principal , 1956 .

[29]  On families of large oscillation , 1972 .

[30]  A remark on density characters , 1946 .

[31]  L. Kantorovitch,et al.  Sur les opérations linéaires dans l'espace des fonctions bornées , 1934 .

[32]  M. Katětov.  Products of filters , 1968 .

[33]  M. Katětov.  On descriptive classification of functions , 1972 .

[34]  Z. Frolík.  Maps of extremally disconnected spaces, theory of types, and applications , 1971 .

[35]  S. Ulam,et al.  Zur Masstheorie in der allgemeinen Mengenlehre , 1930 .

[36]  Leonard Gillman,et al.  Rings of continuous functions , 1961 .

[37]  W. W. Comfort.  A theorem of Stone-Čech type, and a theorem of Tychonoff type, without the axiom of choice; and their realcompact analogues , 1968 .

[38]  W. Wistar Comfort,et al.  Some recent applications of ultrafilters to topology , 1977 .

[39]  V. Saks,et al.  Products of -compact spaces , 1971 .

[40]  H. J. Keisler,et al.  From Accessible to Inaccessible Cardinals , 1967 .

[41]  Neil Hindman.  The existence of certain ultra-filters on and a conjecture of Graham and Rothschild , 1972 .

[42]  Robert Ellis,et al.  Lectures in Topological Dynamics , 1969 .

[43]  Kenneth Kunen.  Some points in N , 1976 .

[44]  R. Woods,et al.  Products of sequentially compact spaces and the $V$-process , 1977 .

[45]  W. Rudin.  Homogeneity Problems in the Theory of Čech Compactifications , 1956 .

[46]  Aimo Tietäväinen.  Proof of a conjecture of S. Chowla , 1975 .

[47]  Характеры и типы точечных множеств , 1962 .

[48]  Stanisław Ulam.  Concerning functions of sets , 1929 .

[49]  Ryszard Engelking.  Zarys topologii ogólnej , 1965 .

[50]  Mary Ellen Rudin,et al.  Partial orders on the types in , 1971 .

[51]  Bedřich Pospíšil,et al.  Remark on Bicompact Spaces , 1937 .

[52]  Frederick Rowbottom,et al.  Some strong axioms of infinity incompatible with the axiom of constructibility , 1971 .

[53]  S. Negrepontis.  The existence of certain uniform ultrafilters , 1969 .

[54]  W. Wistar Comfort,et al.  The Theory of Ultrafilters , 1974 .

[55]  Paul Erdös,et al.  A colour problem for infinite graphs and a problem in the theory of relations , 1951 .

[56]  Kenneth Kunen,et al.  Ultrafilters and independent sets , 1972 .

[57]  M. Stone.  Applications of the theory of Boolean rings to general topology , 1937 .