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Transactions of the American Mathematical Society

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On topological classification of function spaces $ C\sb p(X)$ of low Borel complexity


Authors: T. Dobrowolski, W. Marciszewski and J. Mogilski
Journal: Trans. Amer. Math. Soc. 328 (1991), 307-324
MSC: Primary 54C35; Secondary 57N17, 57N20
DOI: https://doi.org/10.1090/S0002-9947-1991-1065602-X
MathSciNet review: 1065602
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Abstract: We prove that if $ X$ is a countable nondiscrete completely regular space such that the function space $ {C_p}(X)$ is an absolute $ {F_{\sigma \,\delta }}$-set, then $ {C_p}(X)$ is homeomorphic to $ {\sigma ^\infty }$, where $ \sigma = \{ ({x_i}) \in {{\mathbf{R}}^\infty }:{x_i}= 0$ for all but finitely many $ i\} $. As an application we answer in the negative some problems of A. V. Arhangel'skii by giving examples of countable completely regular spaces $ X$ and $ Y$ such that $ X$ fails to be a $ {b_R}$-space and a $ k$-space (and hence $ X$ is not a $ {k_\omega }$-space and not a sequential space) and $ Y$ fails to be an $ {\aleph _0}$-space while the function spaces $ {C_p}(X)$ and $ {C_p}(Y)$ are homeomorphic to $ {C_p}(\mathfrak{X})$ for the compact metric space $ \mathfrak{X}= \{ 0\} \cup \{ {n^{ - 1}}:n= 1,2, \ldots \} $.


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  • [AU] P. S. Aleksandrov and P. S. Urysohn, Memuar o kompaktnych topologičeskich prostranstvach, 3rd ed., Moscow, 1971.
  • [Ar$ _{1}$] A. V. Arhangel'skii, A survey of $ {C_p}$-theory, Q & A in General Topology 5 (1987), special issue. MR 909494 (89c:54031)
  • [Ar$ _{2}$] -, Some results and problems in $ {C_p}(X)$-theory, Proc. Sixth Prague Topological Sympos., 1988, pp. 11-31.
  • [BGvM] J. Baars, J. de Groot and J. van Mill, Topological equivalence of certain function spaces. II, VU (Amsterdam) rapportur. 321, December 1986.
  • [BGvMP] J. Baars, J. de Groot, J. van Mill, and J. Pelant, On topological and linear homeomorphisms of certain function spaces, Topology Appl. 32 (1989), 267-277. MR 1007105 (91b:54036)
  • [BM] M. Bestvina and J. Mogilski, Characterizing certain incomplete infinite-dimensional absolute retracts, Michigan Math. J. 33 (1986), 291-313. MR 856522 (88f:57023)
  • [Ca$ _{1}$] J. Calbrix, Classes de Baire et espaces d'applications continues, C. R. Acad. Sci. Paris 301 (1985), 759-762. MR 817590 (87a:54021)
  • [Ca$ _{2}$] -, Filtres boréliens sur l'ensemble des entiers et espaces des applications continues, Rev. Roumaine Math. Pures Appl. 33 (1988), 655-661. MR 962412 (90a:54043)
  • [DGLvM] J. Dijkstra, T. Grilliot, D. Lutzer, and J. van Mill, Function spaces of low Borel complexity, Proc. Amer. Math. Soc. 94 (1985), 703-710. MR 792287 (87a:54018)
  • [DGM] T. Dobrowolski, S. P. Gulko and J. Mogilski, Function spaces homeomorphic to the countable product of $ l_2^f$, Topology Appl. 34 (1990), 153-160. MR 1041769 (91c:57024)
  • [DM] T. Dobrowolski and J. Mogilski, Certain sequence and function spaces homeomorphic to the countable product of $ l_2^f$, J. London Math. Soc. (to appear).
  • [Ku] K. Kuratowski, Topology, vol. I, Academic Press, New York, 1966. MR 0217751 (36:840)
  • [LM] D. Lutzer and R. McCoy, Category in function spaces. I, Pacific J. Math. 90 (1980), 145-168. MR 599327 (82k:54019)
  • [LvMP] D. Lutzer, J. van Mill, and R. Pol, Descriptive complexity of function spaces, Trans. Amer. Math. Soc. 291 (1985), 121-128. MR 797049 (87e:54046)
  • [vM] J. van Mill, Topological equivalence of certain function spaces, Comp. Math. 63 (1987), 159-188. MR 906368 (89b:54020)
  • [SR] J. Saint Raymond, Fonctions boréliennes sur quotient, Bull. Sci. Math. (2) 100 (1976), 141-147. MR 0460578 (57:571)
  • [Ta] M. Talagrand, Compacts de fonctions mesurables et filtres non mesurables, Studia Math. 67 (1980), 13-43. MR 579439 (82e:28009)
  • [T] H. Torurńczyk, Concerning locally homotopy negligible sets and characterization of $ {l_2}$-manifolds, Fund. Math. 101 (1978), 93-110. MR 518344 (80g:57019)
  • [V] V. S. Varadarajan, Measures on topological spaces, Mat. Sb. 55 (97) (1961), 35-100; English transl., Amer. Math. Soc. Transl. (2) 48 (1965), 161-228. MR 0148838 (26:6342)
  • [W] W. W. Wadge, Degrees of complexity of subsets of the Baire space, Notices Amer. Math. Soc. 1972, A-714.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-1065602-X
Keywords: Function space, pointwise convergence topology, Borelian filter, $ k$-space, $ {k_\omega }$-space, $ {b_R}$-space, $ {\aleph _0}$-space
Article copyright: © Copyright 1991 American Mathematical Society

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