Ultrafilters on the collection of compact subsets of a topological space

Document Type : Original Article

Authors

Department of Mathematics, University of Guilan, Rasht, Guilan, Iran

Abstract

Let $X$ be a Hausdorff topological space and let $\I=\fc(X)$ denote the collection of all non empty compact subsets of $X$. Let $\beta I$ denote the collection of all ultrafilters on the set $I$ and let $(\beta \I,\uplus)$ be the compact Hausdorff right topological semigroup that is the Stone-$\check{C}$ech compactification of the semigroup $(\I, \cup)$ equipped with discrete topology. A. D. Grainger and S. Koppelberg  considered $(\beta \I,\uplus)$ when $X$ is a discrete space and obtained some results about it. In this paper we study $(\beta \I,\uplus)$ when $X$ is a topological space and obtain some results from [2,3,6] .

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References

[1] Berglund, J. F., H. D. Junghenn & P. Milnes. (1989). Analysis on Semigroups: Function Spaces, Compactifications, Representations, Wiley, New York.
[2] Grainger, A.D. (2003). Ultrafilters on the collection of finite subsets of an infinite set. Semigroup Forum, 67, 443–453. DOI: https://doi.org/10.1007/s00233-002-0001-9.
[3] Grainger, A.D. (2006). Ideals of ultrafilters on the collection of finite subsets of an infinite set. Semigroup Forum, 73, 234–242. DOI: https://doi.org/10.1007/s00233-006-0632-3.
[4] Grainger, A.D. (2008). Remarks on Ultrafilters on the Collection of Finite Subsets of an Infinite set.
Contemporary Mathematics, 467, 73–84.
[5] Hindman, N., & D. Strauss. (2011). Algebra in the Stone-Cech Compactification, Theory and Application. Springer Series in Computational Mathematics, Walter de Gruyter, Berlin.
[6] Koppelberg, S. (2006). The Stone-Cˇech Compactification of Semilattice. Semigroup Forum, 72, 63–74. 
Measure Algebras and Applications
Volume 2, Issue 2 - Serial Number 3
December 2024
Pages 151-161

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History

  • Receive Date: 11 September 2024
  • Revise Date: 24 November 2024
  • Accept Date: 17 December 2024

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