P-embedding and product spaces

Morita, Kiiti; Hoshina, Takao

doi:10.4064/fm-93-1-71-80

Cited by 27 publications

(4 citation statements)

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“…To prove the first statement, suppose on the contrary that there exists a locally finite family {G n : n ∈ N} of non-empty open sets in X such that G n ⊆ U(f, n) \ F for each n ∈ N. Take a non-empty zero-set Z n and a cozero-set W n in X with Z n ⊆ W n ⊆ G n for each n ∈ N, and put Z = n∈N Z n . Then, Z is a zero-set in X by [14,Lemma 2.3] and Z ∩ F = ∅. Since Z ∩ U(f, n) = ∅ for each n ∈ N, this contradicts the fact that f satisfies ( * ).…”

Section: Full Zero-setsmentioning

confidence: 73%

Compactness of certain bounded zero-sets in completely regular spaces

Mou

2005

Topology and its Applications

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Let X be a completely regular T 1 -space. A zero-set Z in X is called a full zero-set if cl βX Z is a zero-set in theČech-Stone compactification βX of X. As an generalization of theorems by W.G. McArthur and V. V. Uspenskii, we prove that every bounded, full zero-set F in X is compact if either (i) X has a regular G δ -diagonal or (ii) X is a Baire space such that every open cover has a σ -point-finite open refinement. In case (i), F is metrizable by Šneǐder's theorem. We also apply this to show that if the Dieudonné completion µX of X is a paracompact M-space, then X is metrizable if either (i) or (iii) X is a Baire space with a σ -point-finite base.

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Section: Full Zero-setsmentioning

confidence: 73%

Compactness of certain bounded zero-sets in completely regular spaces

Mou

2005

Topology and its Applications

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“…Next, according to [35,Lemma 2.3], the set F ¼ S F V : V 2 V f g is a zero-set of X. Then, in this case, we may set…”

Section: Extensions By Means Of Expansions and Selectionsmentioning

confidence: 99%

Extensions by Means of Expansions and Selections

2006

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We provide proper mapping-characterizations of some embedding-like properties weaker than P ! -embedding. For instance, we show that a subset A of a space X is U ! -embedded in X if and only if for every continuous map g: A ! Y into a Banach space Y of weight w(Y) r !, there exists a continuous set-valued mapping ' of X into the nonempty compact subsets of Y such that g is a selection for ' j A (i.e., g(x) 2 '(x) for every x 2 A). On the other hand, we show that a subset A is C*-embedded in X if and only if for every continuous set-valued mapping ' of X into the non-empty compact subsets of a Banach space Y, every continuous selection g: A ! Y for ' j A is continuously extendable to the whole of X. Combining both results we get the well-known mapping-characterization of P ! -embedding which makes more transparent the relation FP ! ¼ U ! þ C*_. Other weak components of P ! -embedding are described in terms of expansions and selections, possible applications are demonstrated as well. (2000): 54C20, 54C25, 54C45, 54C60, 54C65. Mathematics Subject Classifications

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“…Also X X C" is P-embedded in X X Y by Theorem 5.2 in [16] and Theorem 4 in [8]. Thus, by Theorem 2.4 in [9], there is a locally finite family {H{ n\X g A} of cozero-sets of X X Y such that Ax n(Xx C") c H'Xn n(Xx C") c/M foreachX g A. Let #x " be open sets of X X Y such that Ax n (X X C") c #x ", Cl(#x ") C ÄJ ".…”

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confidence: 88%

On Collectionwise Normality of Product Spaces. II

Chiba¹

1984

Proceedings of the American Mathematical Society

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In this paper the following results will be obtained. (1) Let X be a closed image of a normal M-space and Y a paracompact first countable /"-space. If XX 7 is normal, then X X Y is collectionwise normal.(2) Let X be a collectionwise normal space and Y a a-locally compact paracompact space. If X X Y is normal, then Y x y is collectionwise normal.Theorem 2. Let X be a collectionwise normal space and Y a o-locally compact paracompact space and X X Y normal. Then XX Y is collectionwise normal.The definitions of M-spaces and P-spaces are due to Morita [7] and o-spaces are due to Okuyama [12].

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P-embedding and product spaces

Cited by 27 publications

References 0 publications

Compactness of certain bounded zero-sets in completely regular spaces

Compactness of certain bounded zero-sets in completely regular spaces

Extensions by Means of Expansions and Selections

On Collectionwise Normality of Product Spaces. II

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