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On the Lindelöf Property of Products of Spaces C

p

( X )

Oleg Okunev

Benemérita Universidad Autónoma de Puebla

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All spaces are assumed to be Tychonoff.

C_p(X,Z) ={f :f is a continuous function fromX toZ}with the topology of pointwise convergence.

C_p(X,Z)⊂Z^X. C_p(X,R) =C_p(X)

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Problem(A.V. Arhangel’skii, 198?)Let X be a space such that C_p(X)is Lindelöf. Must C_p(X)×C_p(X)be Lindelöf? What if X is compact?

C_p(X)×C_p(Y) =C_p(X ⊕Y)

C_p(X)×C_p(X) =C_p(X ⊕X) =C_p(X,R²). C_p(X)always hasccc, so

C_p(X)is Lindelöf ⇐⇒ C_p(X)is paracompact.

Stillunknown:

If G is a Lindelöf topological group, must G×G be Lindelöf?

If G is a paracompact topological group, must G×G be paracompact?

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Stillunknown:

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Stillunknown:

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Stillunknown:

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There are spacesX andY such thatC_p(X)andC_p(Y)are Lindelöf, butC_p(X)×C_p(Y)is not.

First example by A. Leiderman and V. Malykhin (1988);

bothX andY with one non-isolated point, forcing.

K. Tamano, OO (1996):There are a separable, scattered,

σ-compact X and a countable Y such that C_p(X)is Lindelöf, and C_p(X)×C_p(Y)is not.

w(C_p(Y)) =|Y|,

so in this exampleC_p(Y)is second-countable.

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There are spacesX andY such thatC_p(X)andC_p(Y)are Lindelöf, butC_p(X)×C_p(Y)is not.

First example by A. Leiderman and V. Malykhin (1988);

bothX andY with one non-isolated point, forcing.

K. Tamano, OO (1996):There are a separable, scattered,

σ-compact X and a countable Y such that C_p(X)is Lindelöf, and C_p(X)×C_p(Y)is not.

w(C_p(Y)) =|Y|,

so in this exampleC_p(Y)is second-countable.

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Stillunknown:

LetX andY be compact. SupposeC_p(X)andC_p(Y)are Lindelöf.

MustC_p(X)×C_p(Y)be Lindelöf?

LetX be compact andY countable. IfC_p(X)is Lindelöf, must C_p(X)×C_p(Y)be Lindelöf?

IfC_p(X)is Lindelöf, mustC_p(X)×ω^ωbe Lindelöf?

(equivalent: ifC_p(X)is Lindelöf, mustC_p(X ⊕ω)be Lindelöf?)

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Stillunknown:

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Stillunknown:

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Theorem(A. Arhangel’skii, E. Reznichenko, 198?).Let X be compact zero-dimensional. If C_p(X)is Lindelöf, then C_p(X)^ω is Lindelöf.

Theorem(OO, 2011). IfdimX =0and Z is a locally compact second-countable space, then C_p(X,Z)is a continuous image of a closed subspace of C_p(X).

C_p(X)ⁿ =C_p(X,Rⁿ), so

Corllary(OO, 2011). If dimX =0and C_p(X)is Lindelöf, then for every n∈ω, C_p(X)ⁿ is Lindelöf.

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Open questions.

IfdimX =0and C_p(X)is Lindelöf, must C_p(X)^ω be Lindelöf?

IfindX =0and C_p(X)is Lindelöf, must C_p(X)×C_p(X)be Lindelöf?

IfindX =0and C_p(X)is Lindelöf, mustdimX =0?

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