A contribution to the topological classification of the spaces Ср(X)

Cauty, Robert; Dobrowolski, Tadeusz; Marciszewski, Witold

Cauty, Robert ; Dobrowolski, Tadeusz ; Marciszewski, Witold

Fundamenta Mathematicae, Tome 142 (1993), p. 269-301 / Harvested from The Polish Digital Mathematics Library

Résumé

We prove that for each countably infinite, regular space X such that $C_{p} (X)$ is a $Z_{σ}$ -space, the topology of $C_{p} (X)$ is determined by the class $F_{0} (C_{p} (X))$ of spaces embeddable onto closed subsets of $C_{p} (X)$ . We show that $C_{p} (X)$ , whenever Borel, is of an exact multiplicative class; it is homeomorphic to the absorbing set $Ω_{α}$ for the multiplicative Borel class $M_{α}$ if $F_{0} (C_{p} (X)) = M_{α}$ . For each ordinal α ≥ 2, we provide an example $X_{α}$ such that $C_{p} (X_{α})$ is homeomorphic to $Ω_{α}$ .

Publié le : 1993-01-01

Zbl 0813.54009

EUDML-ID : urn:eudml:doc:211987

@article{bwmeta1.element.bwnjournal-article-fmv142i3p269bwm,
     author = {Robert Cauty and Tadeusz Dobrowolski and Witold Marciszewski},
     title = {A contribution to the topological classification of the spaces Sr(X)},
     journal = {Fundamenta Mathematicae},
     volume = {142},
     year = {1993},
     pages = {269-301},
     zbl = {0813.54009},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv142i3p269bwm}
}

Cauty, Robert; Dobrowolski, Tadeusz; Marciszewski, Witold. A contribution to the topological classification of the spaces Ср(X). Fundamenta Mathematicae, Tome 142 (1993) pp. 269-301. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv142i3p269bwm/

Bibliographie

[00000] [1] C. Bessaga and A. Pełczyński, Selected Topics in Infinite-Dimensional Topology, PWN, Warszawa 1975.

[00001] [2] M. Bestvina and J. Mogilski, Characterizing certain incomplete infinite-dimensional retracts, Michigan Math. J. 33 (1986), 291-313. | Zbl 0629.54011

[00002] [3] J. Calbrix, Classes de Baire et espaces d'applications continues, C. R. Acad. Sci. Paris 301 (1985), 759-762. | Zbl 0581.54023

[00003] [4] J. Calbrix, Filtres boréliens sur l'ensemble des entiers et espaces des applications continues, Rev. Roumaine Math. Pures Appl. 33 (1988), 655-661. | Zbl 0659.54009

[00004] [5] R. Cauty, L'espace des fonctions continues d'un espace métrique dénombrable, Proc. Amer. Math. Soc. 113 (1991), 493-501.

[00005] [6] R. Cauty, Sur deux espaces de fonctions non dérivables, preprint.

[00006] [7] R. Cauty, Un exemple d'ensembles absorbants non équivalents, Fund. Math. 140 (1991), 49-61.

[00007] [8] R. Cauty, Ensembles absorbants pour les classes projectives, ibid., to appear.

[00008] [9] R. Cauty and T. Dobrowolski, Applying coordinate products to the topological identification of normed spaces, Trans. Amer. Math. Soc., to appear. | Zbl 0820.57015

[00009] [10] M. M. Choban, Baire sets in complete topological spaces, Ukrain. Math. Zh. 22 (1970), 330-342 (in Russian).

[00010] [11] J. J. Dijkstra, T. Grilliot, D. Lutzer and J. van Mill, Function spaces of low Borel complexity, Proc. Amer. Math. Soc. 94 (1985), 703-710. | Zbl 0525.54010

[00011] [12] J. J. Dijkstra, J. van Mill and J. Mogilski, The space of infinite-dimensional compacta and other topological copies of ${(l_{f})}^{ω}$ , Pacific J. Math. 152 (1992), 255-273. | Zbl 0786.54012

[00012] [13] T. Dobrowolski, S. P. Gulko and J. Mogilski, Function spaces homeomorphic to the countable product of $l_{f}^{2}$ , Topology Appl. 34 (1990), 153-160. | Zbl 0691.57009

[00013] [14] T. Dobrowolski, W. Marciszewski and J. Mogilski, On topological classification of function spaces of low Borel complexity, Trans. Amer. Math. Soc. 328 (1991), 307-324. | Zbl 0768.54016

[00014] [15] T. Dobrowolski and J. Mogilski, Problems on topological classification of incomplete metric spaces, in: Open Problems in Topology, J. van Mill and G. M. Reed (eds.), North-Holland, Amsterdam 1990, 409-429.

[00015] [16] T. Dobrowolski and H. Toruńczyk, Separable complete ANR's admitting a group structure are Hilbert manifolds, Topology Appl. 12 (1981), 229-235. | Zbl 0472.57009

[00016] [17] R. Engelking, General Topology, PWN, Warszawa 1977.

[00017] [18] J. E. Jayne and C. A. Rogers, K-analytic sets, in: Analytic Sets, C. A. Rogers et al. (eds.), Academic Press, London 1980. | Zbl 0524.54028

[00018] [19] K. Kunen and A. W. Miller, Borel and projective sets from the point of view of compact sets, Math. Proc. Cambridge Philos. Soc. 94 (1983), 399-409. | Zbl 0545.54027

[00019] [20] K. Kuratowski, Topology. I, Academic Press, New York 1966.

[00020] [21] L. A. Louveau and J. Saint Raymond, Borel classes and games: Wadge-type and Hurewicz-type results, Trans. Amer. Math. Soc. 304 (1987), 431-467. | Zbl 0655.04001

[00021] [22] D. Lutzer, J. van Mill and R. Pol, Descriptive complexity of function spaces, ibid. 291 (1985), 121-128. | Zbl 0574.54042

[00022] [23] W. Marciszewski, On analytic and coanalytic function spaces $C_{p} (X)$ , Topology Appl., to appear. | Zbl 0785.54020

[00023] [24] J. R. Steel, Analytic sets and Borel isomorphism, Fund. Math. 108 (1980), 83-88. | Zbl 0463.03028

[00024] [25] H. Toruńczyk, Concerning locally homotopy negligible sets and characterization of $l_{2}$ -manifolds, ibid. 101 (1978), 93-110. | Zbl 0406.55003

[00025] [26] W. W. Wadge, Ph.D. thesis, Berkeley 1984.