Non-smooth analysis, optimisation theory and Banach space theory

Borwein, Jonathan M.; Moors, Warren B.

doi:10.1016/b978-044452208-5/50050-8

Cited by 2 publications

(3 citation statements)

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“…To conclude this section, let us mention the following result which gives a description of the class of Namioka spaces and answers in a certain sense Question 1167 (or Question 8.2) in [3]. (1) X is a Namioka space,…”

Section: Remark 42mentioning

confidence: 99%

“…Following [9], a compact space K is said to be co-Namioka if N (X, K) holds for every Baire space X. The class of co-Namioka spaces contains several classes of compact spaces appearing in Banach spaces theory, like Eberlein or Corson compactums ( [11], [10]); in this connection, the reader is referred to [18,22,3] and the references therein for more information. On the other hand, every σ-β-defavorable space (see below) is a Namioka space; this is Christensen-Saint Raymond's theorem [8,24].…”

Section: Introductionmentioning

confidence: 99%

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Some results on separate and joint continuity

Bareche

Bouziad

2010

Topology and its Applications

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Let f : X × K → R be a separately continuous function and C a countable collection of subsets of K. Following a result of Calbrix and Troallic, there is a residual set of points x ∈ X such that f is jointly continuous at each point of {x}×Q, where Q is the set of y ∈ K for which the collection C includes a basis of neighborhoods in K. The particular case when the factor K is second countable was recently extended by Moors and Kenderov to anyČech-complete Lindelöf space K and Lindelöf α-favorable X, improving a generalization of Namioka's theorem obtained by Talagrand. Moors proved the same result when K is a Lindelöf p-space and X is conditionally σ-α-favorable space. Here we add new results of this sort when the factor X is σ C(X) -β-defavorable and when the assumption "base of neighborhoods" in Calbrix-Troallic's result is replaced by a type of countable completeness. The paper also provides further information about the class of Namioka spaces.2000 Mathematics Subject Classification. 54C05,54C35, 54C45, 54C99.

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Section: Remark 42mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Some results on separate and joint continuity

Bareche

Bouziad

2010

Topology and its Applications

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“…The topic here is closely related to the following problem by Talagrand [24] (1985): Let f : X × Y → R be a separately continuous mapping, where X is a Baire space and Y is a compact space; is it true that f admits at least a continuity point in X × Y ? The reader is referred to [5] for further information about this still-open question. According to Corollary 3.8, we have a positive answer if densely many points of Y (or X) are of countable π-character.…”

Section: Introductionmentioning

confidence: 99%

Cliquishness and Quasicontinuity of Two-Variable Maps

Bouziad¹

2013

Can. math. bull.

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Abstract. We study the existence of continuity points for mappings f : X × Y → Z whose x-sections Y ∋ y → f (x, y) ∈ Z are fragmentable and y-sections X ∋ x → f (x, y) ∈ Z are quasicontinuous, where X is a Baire space and Z is a metric space. For the factor Y , we consider two infinite "pointpicking" games G 1 (y) and G 2 (y) defined respectively for each y ∈ Y as follows: in the n-th inning, Player I gives a dense set Dn ⊂ Y , respectively, a dense open set Dn ⊂ Y . Then Player II picks a point yn ∈ Dn; II wins if y is in the closure of {yn : n ∈ N}, otherwise I wins. It is shown that (i) f is cliquish if II has a winning strategy in G 1 (y) for every y ∈ Y , and (ii) f is quasicontinuous if the x-sections of f are continuous and the set of y ∈ Y such that II has a winning strategy in G 2 (y) is dense in Y . Item (i) extends substantially a result of Debs and item (ii) indicates that the problem of Talagrand on separately continuous maps has a positive answer for a wide class of "small" compact spaces.

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Non-smooth analysis, optimisation theory and Banach space theory

Cited by 2 publications

References 76 publications

Some results on separate and joint continuity

Some results on separate and joint continuity

Cliquishness and Quasicontinuity of Two-Variable Maps

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