A function space Cp(X) not linearly homeomorphic to Cp(X) × ℝ

Witold Marciszewski

ArticleOriginal scientific text

Title

A function space Cp(X) not linearly homeomorphic to Cp(X) × ℝ

Authors ^1,²

Affiliations

Faculty of Mathematics and Computer Science, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands
Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

Abstract

We construct two examples of infinite spaces X such that there is no continuous linear surjection from the space of continuous functions

c_{p} (X)

onto

c_{p} (X)

× ℝ

. I n p a r t i c \underset{̲}{a} r,

c_p(X)

i s \neg l \in e a r l y h o m e o m or ϕ c \to

c_p(X)

\times ℝ

. One of these examples is compact. This answers some questions of Arkhangel'skiĭ.

Keywords

ENG

function space, pointwise convergence topology,

c_{p} (X)

, linear homeomorphism

[Ar1] A. V. Arkhangel'skiĭ, Linear homeomorphisms of function spaces, Soviet Math. Dokl. 25 (3) (1982), 852-855.
[Ar2] A. V. Arkhangel'skiĭ, A survey of $C_{p}$ -theory, Questions Answers Gen. Topology 5 (1987), 1-109.
[Ar3] A. V. Arkhangel'skiĭ, Problems in $C_{p}$ -Theory, in: Open Problems in Topology, North-Holland, 1990, 601-615.
[Ar4] A. V. Arkhangel'skiĭ, $C_{p}$ -theory, in: Recent Progress in General Topology, North-Holland, 1992, 1-56.
[BdG] J. Baars and J. de Groot, On Topological and Linear Equivalence of Certain Function Spaces, CWI Tract 86, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1992.
[Ber] Yu. F. Bereznitskiĭ, Nonhomeomorphism between two bicompacta, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 26 (6) (1971), 8-10 (in Russian).
[Be] C. Bessaga, A Lipschitz invariant of normed linear spaces related to the entropy numbers, Rocky Mountain J. Math. 10 (1980), 81-84.
[BPR] C. Bessaga, A. Pełczyński and S. Rolewicz, On diametral approximative dimension and linear homogeneity of F-spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 9 (1961), 677-683.
[Du] E. Dubinsky, Every separable Fréchet space contains a non-stable dense subspace, Studia Math. 40 (1971), 77-79.
[Go] W. T. Gowers, A solution to Banach's hyperplane problem, Bull. London Math. Soc. 26 (1994), 523-530.
[GM] W. T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), 851-874.
[Gu] S. P. Gul'ko, Spaces of continuous functions on ordinals and ultrafilters, Mat. Zametki 47 (4) (1990), 26-34 (in Russian).
[Kun] K. Kunen, Weak P-points in N*, in: Colloq. Math. Soc. János Bolyai 23, North-Holland, 1978, 741-749.
[Ku] K. Kuratowski, Sur la puissance de l'ensemble des "nombres de dimension" au sens de M. Fréchet, Fund. Math. 8 (1926), 201-208.
[KS] K. Kuratowski et W. Sierpiński, Sur un problème de M. Fréchet concernant les dimensions des ensembles linéaires, Fund. Math. 8 (1926), 193-200.
[Ma1] W. Marciszewski, A pre-Hilbert space without any continuous map onto its own square, Bull. Polish Acad. Sci. Math. 31 (1983), 393-397.
[Ma2] W. Marciszewski, A function space C(K) not weakly homeomorphic to C(K) × C(K), Studia Math. 88 (1988), 129-137.
[Ma3] W. Marciszewski, On van Mill's example of a normed X with $X ≉ X \times ℝ$ , preprint.
[vM1] J. van Mill, An introduction to ℝ, in: Handbook of Set-Theoretic Topology, North-Holland, 1984, 503-567.
[vM2] J. van Mill, Domain invariance in infinite-dimensional linear spaces, Proc. Amer. Math. Soc. 101 (1987), 173-180.
[Po1] R. Pol, An infinite-dimensional pre-Hilbert space not homeomorphic to its own square, Proc. Amer. Math. Soc. 90 (1984), 450-454.
[Po2] R. Pol, On metrizable E with $C_{p} (E) ≇ C_{p} (E) \times C_{p} (E)$ , Mathematika 42 (1995), 49-55.
[Ro] S. Rolewicz, An example of a normed space non-isomorphic to its product by the real line, Studia Math. 40 (1971), 71-75.

Title

A function space Cp(X) not linearly homeomorphic to Cp(X) × ℝ

Affiliations

Abstract

Keywords

Bibliography