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Function space topologies deriving from hypertopologies and networks

A. Di Concilio, A. Miranda (2001)

Bollettino dell'Unione Matematica Italiana

In un progetto di generalizzazione delle classiche topologie di tipo «set-open» di Arens-Dugundji introduciamo un metodo generale per produrre topologie in spazi di funzioni mediante l'uso di ipertopologie. Siano X , Y spazi topologici e C ⁢ X , Y l'insieme delle funzioni continue da X verso Y . Fissato un «network» α nel dominio X ed una topologia τ nell'iperspazio C ⁢ L ⁢ Y del codominio Y si genera una topologia τ α in C ⁢ X , Y richiedendo che una rete f λ di C ⁢ X , Y converge in τ α ad f ∈ C ⁢ X , Y se e solo se la rete f λ ⁢ A ¯ converge in τ ad f ⁢ A ¯ ...

Function spaces and local properties

Ziqin Feng, Paul Gartside (2013)

Fundamenta Mathematicae

Necessary conditions and sufficient conditions are given for C p ( X ) to be a (σ-) m₁- or m₃-space. (A space is an m₁-space if each of its points has a closure-preserving local base.) A compact uncountable space K is given with C p ( K ) an m₁-space, which answers questions raised by Dow, Ramírez Martínez and Tkachuk (2010) and Tkachuk (2011).

Functions characterized by images of sets

Krzysztof Ciesielski, Dikran Dikrajan, Stephen Watson (1998)

Colloquium Mathematicae

For non-empty topological spaces X and Y and arbitrary families 𝒜 ⊆ 𝒫 ( X ) and ℬ ⊆ 𝒫 ( Y ) we put 𝒞 𝒜 , ℬ =f ∈ Y X : (∀ A ∈ 𝒜 )(f[A] ∈ ℬ ) . We examine which classes of functions ℱ ⊆ Y X can be represented as 𝒞 𝒜 , ℬ . We are mainly interested in the case when ℱ = 𝒞 ( X , Y ) is the class of all continuous functions from X into Y. We prove that for a non-discrete Tikhonov space X the class ℱ = 𝒞 (X,ℝ) is not equal to 𝒞 𝒜 , ℬ for any 𝒜 ⊆ 𝒫 ( X ) and ℬ ⊆ 𝒫 (ℝ). Thus, 𝒞 (X,ℝ) cannot be characterized by images of sets. We also show that none of the following classes of...

Generalized Helly spaces, continuity of monotone functions, and metrizing maps

Lech Drewnowski, Artur Michalak (2008)

Fundamenta Mathematicae

Given an ordered metric space (in particular, a Banach lattice) E, the generalized Helly space H(E) is the set of all increasing functions from the interval [0,1] to E considered with the topology of pointwise convergence, and E is said to have property (λ) if each of these functions has only countably many points of discontinuity. The main objective of the paper is to study those ordered metric spaces C(K,E), where K is a compact space, that have property (λ). In doing so, the guiding idea comes...

Krasinkiewicz maps from compacta to polyhedra

Eiichi Matsuhashi (2006)

Bulletin of the Polish Academy of Sciences. Mathematics

We prove that the set of all Krasinkiewicz maps from a compact metric space to a polyhedron (or a 1-dimensional locally connected continuum, or an n-dimensional Menger manifold, n ≥ 1) is a dense G δ -subset of the space of all maps. We also investigate the existence of surjective Krasinkiewicz maps from continua to polyhedra.

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