Countably compact first countable subspaces of ordinals have the Sokolov property

A space X is Sokolov if for any sequence {F n : n ∈ ℕ} where F n is a closed subset of X n for every n ∈ ℕ, there exists a continuous map f : X → X such that nw(f(X)) ≤ ω and f n (F n ) ⊂ F n for all n ∈ ℕ. We prove that if X is a first countable countably compact subspace of an ordinal then X is a...

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Bibliographic Details
Published in:Quaestiones mathematicae 2011-06, Vol.34 (2), p.225-234
Main Author: Tkachuk, Vladimir V.
Format: Article
Language:English
Subjects:
54D06
Secondary: 54D25
countably compact space
D-space
first countable space
function space
iterated function space
Lindelöf space
Primary: 54H11
retraction
Sokolov space
Subspace of an ordinal
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Summary:A space X is Sokolov if for any sequence {F n : n ∈ ℕ} where F n is a closed subset of X n for every n ∈ ℕ, there exists a continuous map f : X → X such that nw(f(X)) ≤ ω and f n (F n ) ⊂ F n for all n ∈ ℕ. We prove that if X is a first countable countably compact subspace of an ordinal then X is a Sokolov space and C P (X) is a D-space; this answers a question of Buzyakova. Thus, for any first countable countably compact subspace X of an ordinal, the iterated function space C p , 2n +1 (X) is Lindelöf for any n ∈ ω Another consequence of the above results is the existence of a first countable Sokolov space of cardinality greater than c.
ISSN:1607-3606
1727-933X
DOI:10.2989/16073606.2011.594237