Selections and sandwich-like properties via semi-continuous Banach-valued functions

Gutev, Valentin; Ohta, Haruto; Yamazaki, Kaori

doi:10.2969/jmsj/1191419128

Cited by 19 publications

(12 citation statements)

References 20 publications

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“…(or u.s.c.) maps coincides with that of Gutev, Ohta and Yamazaki [5] for Y = C 0 (Z) (see [13,Corollary 2.3]). Here, C 0 (Z) is the Banach lattice consisting of all continuous functions s on a topological space Z such that for each ε > 0 the set {z ∈ Z : s(z) ε} is compact, where the linear operations are defined pointwise, s = sup z∈Z s(z) for each s ∈ C 0 (Z), and the order relation is defined as follows: for…”

Section: Terminology Main Theorems and Key Lemmassupporting

confidence: 70%

“…Since g h, from the assumption, there exists f ∈ C(X, Y ) with g f h and [2]). The 'only if' part follows by a quite similar proof to (4) ⇒ (1) of [5,Theorem 4.1].…”

Section: Proofsmentioning

confidence: 85%

“…In the Katětov-Tong insertion theorem, the range 'R' can be replaced by separable Banach lattices c 0 and l p (1 p < ∞), but it cannot be replaced by another separable Banach lattice c ( [5], [8], [13]). So, it seems to be natural to ask whether or not 'R' in the Dowker-Katětov and Michael insertion theorems can be replaced by any non-trivial separable Banach lattice.…”

Section: Introductionmentioning

confidence: 99%

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The range of maps on classical insertion theorems

Yamazaki

2011

Acta Math Hung

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A classical insertion theorem due to Katětov-Tong (or DowkerKatětov, Michael) reads that R can be a test space for the range of maps on the insertion theorem which characterizes the domain to be normal (or normal and countably paracompact, perfectly normal). It is known that the range R in the Katětov-Tong insertion theorem is not necessarily replaced by a non-trivial separable Banach lattice. We show that the range R in the Dowker-Katětov and Michael insertion theorems can be replaced by any non-trivial separable Banach lattice.

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Section: Terminology Main Theorems and Key Lemmassupporting

confidence: 70%

“…Since g h, from the assumption, there exists f ∈ C(X, Y ) with g f h and [2]). The 'only if' part follows by a quite similar proof to (4) ⇒ (1) of [5,Theorem 4.1].…”

Section: Proofsmentioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%

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The range of maps on classical insertion theorems

Yamazaki

2011

Acta Math Hung

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“…because of [13, Theorem 1.2 * ]. Thus, the operator g is the required one in (6) . The proof of the following lemma is straightforward.…”

Section: Locally Bounded Set-valued Mappingsmentioning

confidence: 94%

Locally bounded set-valued mappings and monotone countable paracompactness

Yamazaki

2007

Topology and its Applications

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“…is upper semicontinuous if for every x 2 X and every " > 0, there exists a neighbourhood G of x in X such that if x 0 2 G, then g(x 0 )() < g(x)() þ " for every 2 c 0 (! ), see [20].…”

Section: Embedding Properties and Expansionsmentioning

confidence: 97%

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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Selections and sandwich-like properties via semi-continuous Banach-valued functions

Cited by 19 publications

References 20 publications

The range of maps on classical insertion theorems

The range of maps on classical insertion theorems

Locally bounded set-valued mappings and monotone countable paracompactness

Extensions by Means of Expansions and Selections

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