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

Robert Cauty; Tadeusz Dobrowolski; Witold Marciszewski

Fundamenta Mathematicae (1993)

  • Volume: 142, Issue: 3, page 269-301
  • ISSN: 0016-2736

Abstract

top
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 .

How to cite

top

Cauty, Robert, Dobrowolski, Tadeusz, and Marciszewski, Witold. "A contribution to the topological classification of the spaces Ср(X)." Fundamenta Mathematicae 142.3 (1993): 269-301. <http://eudml.org/doc/211987>.

@article{Cauty1993,
abstract = {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 $Ω_α$.},
author = {Cauty, Robert, Dobrowolski, Tadeusz, Marciszewski, Witold},
journal = {Fundamenta Mathematicae},
keywords = {function space; pointwise convergence topology; absorbing sets; Borel and projective filters; Borel filters; projective filters; strong -universality; topology of pointwise convergence; absorbing set method},
language = {eng},
number = {3},
pages = {269-301},
title = {A contribution to the topological classification of the spaces Ср(X)},
url = {http://eudml.org/doc/211987},
volume = {142},
year = {1993},
}

TY - JOUR
AU - Cauty, Robert
AU - Dobrowolski, Tadeusz
AU - Marciszewski, Witold
TI - A contribution to the topological classification of the spaces Ср(X)
JO - Fundamenta Mathematicae
PY - 1993
VL - 142
IS - 3
SP - 269
EP - 301
AB - 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 $Ω_α$.
LA - eng
KW - function space; pointwise convergence topology; absorbing sets; Borel and projective filters; Borel filters; projective filters; strong -universality; topology of pointwise convergence; absorbing set method
UR - http://eudml.org/doc/211987
ER -

References

top
  1. [1] C. Bessaga and A. Pełczyński, Selected Topics in Infinite-Dimensional Topology, PWN, Warszawa 1975. 
  2. [2] M. Bestvina and J. Mogilski, Characterizing certain incomplete infinite-dimensional retracts, Michigan Math. J. 33 (1986), 291-313. Zbl0629.54011
  3. [3] J. Calbrix, Classes de Baire et espaces d'applications continues, C. R. Acad. Sci. Paris 301 (1985), 759-762. Zbl0581.54023
  4. [4] J. Calbrix, Filtres boréliens sur l'ensemble des entiers et espaces des applications continues, Rev. Roumaine Math. Pures Appl. 33 (1988), 655-661. Zbl0659.54009
  5. [5] R. Cauty, L'espace des fonctions continues d'un espace métrique dénombrable, Proc. Amer. Math. Soc. 113 (1991), 493-501. 
  6. [6] R. Cauty, Sur deux espaces de fonctions non dérivables, preprint. 
  7. [7] R. Cauty, Un exemple d'ensembles absorbants non équivalents, Fund. Math. 140 (1991), 49-61. 
  8. [8] R. Cauty, Ensembles absorbants pour les classes projectives, ibid., to appear. 
  9. [9] R. Cauty and T. Dobrowolski, Applying coordinate products to the topological identification of normed spaces, Trans. Amer. Math. Soc., to appear. Zbl0820.57015
  10. [10] M. M. Choban, Baire sets in complete topological spaces, Ukrain. Math. Zh. 22 (1970), 330-342 (in Russian). 
  11. [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. Zbl0525.54010
  12. [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. Zbl0786.54012
  13. [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. Zbl0691.57009
  14. [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. Zbl0768.54016
  15. [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. 
  16. [16] T. Dobrowolski and H. Toruńczyk, Separable complete ANR's admitting a group structure are Hilbert manifolds, Topology Appl. 12 (1981), 229-235. Zbl0472.57009
  17. [17] R. Engelking, General Topology, PWN, Warszawa 1977. 
  18. [18] J. E. Jayne and C. A. Rogers, K-analytic sets, in: Analytic Sets, C. A. Rogers et al. (eds.), Academic Press, London 1980. Zbl0524.54028
  19. [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. Zbl0545.54027
  20. [20] K. Kuratowski, Topology. I, Academic Press, New York 1966. 
  21. [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. Zbl0655.04001
  22. [22] D. Lutzer, J. van Mill and R. Pol, Descriptive complexity of function spaces, ibid. 291 (1985), 121-128. Zbl0574.54042
  23. [23] W. Marciszewski, On analytic and coanalytic function spaces C p ( X ) , Topology Appl., to appear. Zbl0785.54020
  24. [24] J. R. Steel, Analytic sets and Borel isomorphism, Fund. Math. 108 (1980), 83-88. Zbl0463.03028
  25. [25] H. Toruńczyk, Concerning locally homotopy negligible sets and characterization of l 2 -manifolds, ibid. 101 (1978), 93-110. Zbl0406.55003
  26. [26] W. W. Wadge, Ph.D. thesis, Berkeley 1984. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.