A function space without a condensation onto a -compact space

Author:
Witold Marciszewski

Journal:
Proc. Amer. Math. Soc. **131** (2003), 1965-1969

MSC (2000):
Primary 54C35, 54A10

Published electronically:
October 18, 2002

MathSciNet review:
1955287

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Assuming that the minimal cardinality of a dominating family in is equal to , we construct a subset of a real line such that the space of continuous real-valued functions on does not admit any continuous bijection onto a -compact space. This gives a consistent answer to a question of Arhangel'skii.

**[Ar1]**A. V. Arhangel'skii,*Factorization theorems and function spaces: Stability and monolithicity*, Soviet Math. Dokl.**26**(1982), 177-181.**[Ar2]**-,*Topological Function Spaces*, Kluwer Academic Publishers, Dordrecht, 1992.**[Ar3]**-,*-Theory*, Recent Progress in General Topology (M. Husek and J. van Mill, eds.), Elsevier, 1992, pp. 1-56.**[Ar4]**A. V. Arhangel′skii,*On condensations of 𝐶_{𝑝}-spaces onto compacta*, Proc. Amer. Math. Soc.**128**(2000), no. 6, 1881–1883 (electronic). MR**1751998**, 10.1090/S0002-9939-00-05758-0**[AP]**A. V. Arkhangel′skiĭ and V. I. Ponomarev,*Osnovy obshchei topologii v zadachakh i uprazhneniyakh*, Izdat. “Nauka”, Moscow, 1974 (Russian). MR**0445439****[vD]**Eric K. van Douwen,*The integers and topology*, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 111–167. MR**776622****[En]**Ryszard Engelking,*General topology*, PWN—Polish Scientific Publishers, Warsaw, 1977. Translated from the Polish by the author; Monografie Matematyczne, Tom 60. [Mathematical Monographs, Vol. 60]. MR**0500780****[Ku]**K. Kuratowski,*Topology. Vol. I*, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. MR**0217751****[Mi]**H. Michalewski,*Condensations of projective sets onto compacta*, Proc. Amer. Math. Soc., to appear.**[Py]**E. G. Pytkeev,*The upper bounds of topologies*, Mat. Zametki**20**(1976), no. 4, 489–500 (Russian). MR**0428237**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
54C35,
54A10

Retrieve articles in all journals with MSC (2000): 54C35, 54A10

Additional Information

**Witold Marciszewski**

Affiliation:
Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

Address at time of publication:
Faculty of Sciences, Division of Mathematics and Computer Science, Vrije Universiteit, De Boelelaan $1081^{a}$, 1081 HV Amsterdam, The Netherlands

Email:
wmarcisz@mimuw.edu.pl

DOI:
http://dx.doi.org/10.1090/S0002-9939-02-06668-6

Keywords:
Function space,
pointwise convergence topology,
$C_{p}(X)$,
condensation

Received by editor(s):
July 2, 2001

Received by editor(s) in revised form:
December 4, 2001, and February 8, 2002

Published electronically:
October 18, 2002

Additional Notes:
The author was supported in part by KBN grant 2 P03A 011 15.

Communicated by:
Alan Dow

Article copyright:
© Copyright 2002
American Mathematical Society