Abstract
We prove that if X is a strongly zero-dimensional space, then for every locally compact second-countable space M, C p (X, M) is a continuous image of a closed subspace of C p (X). It follows in particular, that for strongly zero-dimensional spaces X, the Lindelöf number of C p (X)×C p (X) coincides with the Lindelöf number of C p (X). We also prove that l(C p (X n)κ) ≤ l(C p (X)κ) whenever κ is an infinite cardinal and X is a strongly zero-dimensional union of at most κcompact subspaces.
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References
Arhangel’skiĭ A.V., Problems in C p-theory, In: Open Problems in Topology, North-Holland, Amsterdam, 1990, 601–615
Arhangel’skiĭ A.V., Topological Function Spaces, Math. Appl. (Soviet Ser.), 78, Kluwer, Dordrecht, 1992
Engelking R., General Topology, Sigma Ser. Pure Math., 6, Heldermann, Berlin, 1989
Mardešić S., On covering dimension and inverse limits of compact spaces, Illinois J. Math, 1960, 4(2), 278–291
Okunev O., On Lindelöf Σ-spaces of continuous functions in the pointwise topology, Topology Appl., 1993, 49(2), 149–166
Okunev O., Tamano K., Lindelöf powers and products of function spaces, Proc. Amer. Math. Soc., 1996, 124(9), 2905–2916
Tkachuk V.V., Some criteria for C p(X) to be an LΣ(≤ ω)-space (in preparation)
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Okunev, O. The Lindelöf number of C p (X)×C p (X) for strongly zero-dimensional X . centr.eur.j.math. 9, 978–983 (2011). https://doi.org/10.2478/s11533-011-0050-y
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DOI: https://doi.org/10.2478/s11533-011-0050-y