## On the Stone-Čech compactification of the space of closed sets

HTML articles powered by AMS MathViewer

- by John Ginsburg
- Trans. Amer. Math. Soc.
**215**(1976), 301-311 - DOI: https://doi.org/10.1090/S0002-9947-1976-0390992-2
- PDF | Request permission

## Abstract:

For a topological space*X*, we denote by ${2^X}$ the space of closed subsets of

*X*with the finite topology. If

*X*is normal and ${T_1}$, the map $F \to {\text {cl}_{\beta X}}F$ is an embedding of ${2^X}$ onto a dense subspace of ${2^{\beta X}}$, and, in this way, we regard ${2^{\beta X}}$ as a compactification of ${2^X}$. This paper is motivated by the following question. When can ${2^{\beta X}}$ be identified as the Stone-Čech compactification of ${2^X}$? In [11], J. Keesling states that $\beta ({2^X}) = {2^{\beta X}}$ implies ${2^X}$ is pseudocompact. We give a proof of this result and establish the following partial converse. If ${2^X} \times {2^X}$ is pseudocompact, then $\beta ({2^X}) = {2^{\beta X}}$. A corollary of this theorem is that $\beta ({2^X}) = {2^{\beta X}}$ when

*X*is ${\aleph _0}$-bounded.

## References

- Robert L. Blair and Anthony W. Hager,
*Extensions of zero-sets and of real-valued functions*, Math. Z.**136**(1974), 41–52. MR**385793**, DOI 10.1007/BF01189255 - W. W. Comfort and Kenneth A. Ross,
*Pseudocompactness and uniform continuity in topological groups*, Pacific J. Math.**16**(1966), 483–496. MR**207886**, DOI 10.2140/pjm.1966.16.483 - 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,
*Some results on the countable compactness and pseudocompactness of hyperspaces*, Canadian J. Math.**27**(1975), no. 6, 1392–1399. MR**400145**, DOI 10.4153/CJM-1975-142-9 - Irving Glicksberg,
*Stone-Čech compactifications of products*, Trans. Amer. Math. Soc.**90**(1959), 369–382. MR**105667**, DOI 10.1090/S0002-9947-1959-0105667-4 - Anthony W. Hager,
*Some remarks on the tensor product of function rings*, Math. Z.**92**(1966), 210–224. MR**193613**, DOI 10.1007/BF01111186 - Karl Heinrich Hofmann and Paul S. Mostert,
*Elements of compact semigroups*, Charles E. Merrill Books, Inc., Columbus, Ohio, 1966. MR**0209387** - James Keesling,
*Normality and properties related to compactness in hyperspaces*, Proc. Amer. Math. Soc.**24**(1970), 760–766. MR**253292**, DOI 10.1090/S0002-9939-1970-0253292-7 - James Keesling,
*Normality and compactness are equivalent in hyperspaces*, Bull. Amer. Math. Soc.**76**(1970), 618–619. MR**254812**, DOI 10.1090/S0002-9904-1970-12459-4 - James Keesling,
*On the equivalence of normality and compactness in hyperspaces*, Pacific J. Math.**33**(1970), 657–667. MR**267516**, DOI 10.2140/pjm.1970.33.657
—, - K. Kuratowski,
*Topology. Vol. I*, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR**0217751** - Jimmie D. Lawson,
*Intrinsic topologies in topological lattices and semilattices*, Pacific J. Math.**44**(1973), 593–602. MR**318031**, DOI 10.2140/pjm.1973.44.593 - Ernest Michael,
*Topologies on spaces of subsets*, Trans. Amer. Math. Soc.**71**(1951), 152–182. MR**42109**, DOI 10.1090/S0002-9947-1951-0042109-4 - Kiiti Morita,
*Completion of hyperspaces of compact subsets and topological completion of open-closed maps*, General Topology and Appl.**4**(1974), 217–233. MR**350683**, DOI 10.1016/0016-660X(74)90023-3 - Dona Papert Strauss,
*Topological lattices*, Proc. London Math. Soc. (3)**18**(1968), 217–230. MR**227948**, DOI 10.1112/plms/s3-18.2.217 - Phillip Zenor,
*On the completeness of the space of compact subsets*, Proc. Amer. Math. Soc.**26**(1970), 190–192. MR**261538**, DOI 10.1090/S0002-9939-1970-0261538-4

*Compactness related properties in hyperspaces*, Lecture Notes in Math., vol. 171, Springer-Verlag, Berlin and New York, 1970.

## Bibliographic Information

- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**215**(1976), 301-311 - MSC: Primary 54B20
- DOI: https://doi.org/10.1090/S0002-9947-1976-0390992-2
- MathSciNet review: 0390992