Condensations of Cp(X) onto σ-compact spaces


  • Vladimir V. Tkachuk Universidad Autónoma Metropolitana



Condensation, Continuous image, Lindelöf Σ-space, σ –compact space, Topology of pointwise convergence, Network weight, Tightness, Lindelöf space


We show, in particular, that if nw(Nt) ≤  k for any t ϵ  T and C is a dense subspace of the product ǁ{Nt : t 2 T} then, for any continuous (not necessarily surjective) map ϕ : C_K of C into a compact space K with t(K) ≤ k, we have ψ(ϕ (C)) ≤ k. This result has several applications in Cp-theory. We prove, among other things, that if K is a non-metrizable Corson compact space then Cp(K) cannot be condensed onto a σ-compact space. This answers two questions published by Arhangel’skii and Pavlov.


Download data is not yet available.

Author Biography

Vladimir V. Tkachuk, Universidad Autónoma Metropolitana

Departamento de Matemáticas


A. V. Arhangel’skii, The structure and classification of topological spaces and cardinal invariants (in Russian), Uspehi Mat. Nauk 33, no. 6 (1978), 29–84.

A. V. Arhangel’skii, On relationship between invariants of topological groups and their subspaces (in Russian), Uspehi Mat. Nauk 35, no. 3 (1980), 3–22.

A. V. Arhangel’skii, Topological Function Spaces, Kluwer Academic Publishers, Mathematics and Its Applications 78, Dordrecht 1992.

A. V. Arhangel’skii and O.I. Pavlov, A note on condensations of Cp(X) onto compacta, (in Russian), Comment. Math. Univ. Carolinae 43, no. 3 (2002), 485–492.

F. Casarrubias–Segura, On compact weaker topologies in function spaces, Topology Appl. 115 (2001), 291–298.

R. Engelking, General Topology, PWN, Warszawa, 1977.

W. Marciszewski, A function space Cp(X) without a condensation onto a -compact space, Proc. Amer. Math. Soc. 131, no. 6 (2003), 1965–1969.

O. G. Okunev, On Lindelöf -spaces of continuous functions in the pointwise topology, Topology Appl. 49, no. 2 (1993), 149-166.

E. G. Pytkeev, Upper bounds of topologies, Math. Notes 20, no. 4 (1976), 831–837.

B. E. Shapirovsky, Cardinal invariants in compact spaces (in Russian), Seminar Gen. Topol., ed. by P.S. Alexandroff, Moscow Univ. P.H., Moscow, 1981, 162–187.

L. V. Shirokov, On bicompacta which are continuous images of dense subspaces of topological products (in Russian), Soviet Institute of Science and Technology Information (VINITI), Registration number 2946–81, Moscow, 1981.

M. G. Tkachenko, Factorization theorems for topological groups and their applications, Topology Appl. 38 (1991), 21–37.

V. V. Tkachuk, Decomposition of Cp(X) into a countable union of subspaces with ”good” properties implies ”good” properties of Cp(X), Trans. Moscow Math. Soc. 55 (1994), 239–248.

V. V. Tkachuk, Behaviour of the Lindel¨of -property in iterated function spaces, Topology Appl. 107 (2000), 297–305.


How to Cite

V. V. Tkachuk, “Condensations of Cp(X) onto σ-compact spaces”, Appl. Gen. Topol., vol. 10, no. 1, pp. 39–48, Apr. 2009.



Regular Articles