Continuous selections avoiding extreme points

Yamauchi, Takamitsu

doi:10.1016/j.topol.2007.02.016

Cited by 6 publications

(4 citation statements)

References 7 publications

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“…Indeed, the 'if' part follows from [1, Theorem 2.1] and [12,Corollary 11]. For the 'only if' part, X is normal and κ-paracompact by Fact 1, and X is perfect by Lemma 2.5.…”

Section: Proofsmentioning

confidence: 97%

“…Indeed, the 'if' part follows from [1, Theorem 2.1] and [12,Theorem 8]. For the 'only if' part, by Lemma 2.4, X is normal.…”

Section: Proofsmentioning

confidence: 98%

“…In our formulation, we mainly concern the 'only if' part, because the 'if' part is immediately obtained by a consequence of known selection theorems [1], [2], [7], [12]. In this viewpoint, the following are key lemmas in our research.…”

Section: And Only If X Is Perfectly Normalmentioning

confidence: 99%

“…(2) The 'if' part follows from [1,Corollary 2.3] and [12,Theorem 2]. The 'only if' part: Assume that every pair of maps g ∈ USC(A, Y ) and…”

Section: Proofsmentioning

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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“…Indeed, the 'if' part follows from [1, Theorem 2.1] and [12,Corollary 11]. For the 'only if' part, X is normal and κ-paracompact by Fact 1, and X is perfect by Lemma 2.5.…”

Section: Proofsmentioning

confidence: 97%

“…Indeed, the 'if' part follows from [1, Theorem 2.1] and [12,Theorem 8]. For the 'only if' part, by Lemma 2.4, X is normal.…”

Section: Proofsmentioning

confidence: 98%

Section: And Only If X Is Perfectly Normalmentioning

confidence: 99%

“…(2) The 'if' part follows from [1,Corollary 2.3] and [12,Theorem 2]. The 'only if' part: Assume that every pair of maps g ∈ USC(A, Y ) and…”

Section: Proofsmentioning

confidence: 99%

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The range of maps on classical insertion theorems

Yamazaki

2011

Acta Math Hung

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Finite-to-one maps into Euclidean manifolds and spaces with disjoint disks properties

Karassev

Tuncali

Valov

2010

Topology and its Applications

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Embeddings of finite-dimensional compacta in Euclidean spaces

Bogataya

Богатый

Valov

2012

Topology and its Applications

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If g is a map from a space X into R m and q is an integer, let B q,d,m (g) be the set of all planes Π d ⊂ R m such that |g −1 (Π d )| ≥ q. Let also H(q, d, m, k) denote the maps g : X → R m such that dim B q,d,m (g) ≤ k. We prove that for any n-dimensional metric compactum X each of the sets H(3, 1, m, 3n + 1 − m) and H(2, 1, m, 2n) is dense and G δ in the function space C(X, R m ) provided m ≥ 2n + 1 (in this case H(3, 1, m, 3n + 1 − m) and H(2, 1, m, 2n) can consist of embeddings). The same is true for the sets H(1, d, m, n + d(m − d)) ⊂ C(X, R m ) if m ≥ n + d, and H(4, 1, 3, 0) ⊂ C(X, R 3 ) if dim X ≤ 1. This results complements an authors' result from [5]. A parametric version of the above theorem, as well as a partial answer of a question from [4] and [5] are also provided.

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Continuous selections avoiding extreme points

Cited by 6 publications

References 7 publications

The range of maps on classical insertion theorems

The range of maps on classical insertion theorems

Finite-to-one maps into Euclidean manifolds and spaces with disjoint disks properties

Embeddings of finite-dimensional compacta in Euclidean spaces

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