Is there a point of 𝜔* that sees all others?

Hindman, Neil

doi:10.1090/S0002-9939-1988-0931732-9

Is there a point of $\omega ^ *$ that sees all others?
HTML articles powered by AMS MathViewer

by Neil Hindman PDF

Proc. Amer. Math. Soc. 104 (1988), 1235-1238 Request permission

Abstract:

If the cardinal $c$ of the continuum is singular and $p$ is an ultrafilter on $\omega$ of character $c$, then there is an ultrafilter $q$ on $\omega$ which is not comparable to $p$ in the Rudin-Keisler order.

References

W. W. Comfort, Ultrafilters: an interim report, Surveys in general topology, Academic Press, New York-London-Toronto, Ont., 1980, pp. 33–54. MR 564099
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

Consequences of Martin’s Axiom

K. Kunen, Weak $P$-points in $\textbf {N}^{\ast }$, Topology, Vol. II (Proc. Fourth Colloq., Budapest, 1978) Colloq. Math. Soc. János Bolyai, vol. 23, North-Holland, Amsterdam-New York, 1980, pp. 741–749. MR 588822
Jan van Mill, An introduction to $\beta \omega$, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 503–567. MR 776630
Walter Rudin, Homogeneity problems in the theory of Čech compactifications, Duke Math. J. 23 (1956), 409–419. MR 80902
S. Shelah and M. E. Rudin, Unordered types of ultrafilters, Topology Proc. 3 (1978), no. 1, 199–204 (1979). MR 540490