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Selectors of discrete coarse spaces

Igor Protasov (2022)

Commentationes Mathematicae Universitatis Carolinae

Given a coarse space ( X , ℰ ) with the bornology ℬ of bounded subsets, we extend the coarse structure ℰ from X × X to the natural coarse structure on ( ℬ ∖ { ∅ } ) × ( ℬ ∖ { ∅ } ) and say that a macro-uniform mapping f : ( ℬ ∖ { ∅ } ) → X (or f : [ X ] 2 → X ) is a selector (or 2-selector) of ( X , ℰ ) if f ( A ) ∈ A for each A ∈ ℬ ∖ { ∅ } ( A ∈ [ X ] 2 , respectively). We prove that a discrete coarse space ( X , ℰ ) admits a selector if and only if ( X , ℰ ) admits a 2-selector if and only if there exists a linear order “ ≤ " on X such that the family of intervals { [ a , b ] : a , b ∈ X , a ≤ b } is a base for the bornology ℬ .

Selivanovski hard sets are hard

Janusz Pawlikowski (2015)

Fundamenta Mathematicae

Let H ⊆ Z ⊆ 2 ω . For n ≥ 2, we prove that if Selivanovski measurable functions from 2 ω to Z give as preimages of H all Σₙ¹ subsets of 2 ω , then so do continuous injections.

Semicontinuity and continuous selections for the multivalued superposition operator without assuming growth-type conditions

Hông Thái Nguyêñ (2004)

Studia Mathematica

Let Ω be a measure space, and E, F be separable Banach spaces. Given a multifunction f : Ω × E → 2 F , denote by N f ( x ) the set of all measurable selections of the multifunction f ( · , x ( · ) ) : Ω → 2 F , s ↦ f(s,x(s)), for a function x: Ω → E. First, we obtain new theorems on H-upper/H-lower/lower semicontinuity (without assuming any conditions on the growth of the generating multifunction f(s,u) with respect to u) for the multivalued (Nemytskiĭ) superposition operator N f mapping some open domain G ⊂ X into 2 Y , where X and Y are Köthe-Bochner...

Semicontinuity in L ∞ for polyconvex integrals

Emilio Acerbi, Giuseppe Buttazzo, Nicola Fusco (1982)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Viene studiata la semicontinuità rispetto alla topologia di L ∞ ⁢ ( Ω ; 𝐑 m ) per alcuni funzionali del Calcolo delle Variazioni dipendenti da funzioni a valori vettoriali.

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