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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...

Functor of extension in Hilbert cube and Hilbert space

Piotr Niemiec (2014)

Open Mathematics

It is shown that if Ω = Q or Ω = ℓ 2, then there exists a functor of extension of maps between Z-sets in Ω to mappings of Ω into itself. This functor transforms homeomorphisms into homeomorphisms, thus giving a functorial setting to a well-known theorem of Anderson [Anderson R.D., On topological infinite deficiency, Michigan Math. J., 1967, 14, 365–383]. It also preserves convergence of sequences of mappings, both pointwise and uniform on compact sets, and supremum distances as well as uniform continuity,...

Fundamental pro-groupoids and covering projections

Luis Hernández-Paricio (1998)

Fundamenta Mathematicae

We introduce a new notion of covering projection E → X of a topological space X which reduces to the usual notion if X is locally connected. We use locally constant presheaves and covering reduced sieves to find a pro-groupoid π crs (X) and an induced category pro (π crs (X), Sets) such that for any topological space X the category of covering projections and transformations of X is equivalent to the category pro (π crs (X), Sets). We also prove that the latter category is equivalent to pro (π CX,...

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