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Free non-archimedean topological groups

Michael Megrelishvili, Menachem Shlossberg (2013)

Commentationes Mathematicae Universitatis Carolinae

We study free topological groups defined over uniform spaces in some subclasses of the class 𝐍𝐀 of non-archimedean groups. Our descriptions of the corresponding topologies show that for metrizable uniformities the corresponding free balanced, free abelian and free Boolean 𝐍𝐀 groups are also metrizable. Graev type ultra-metrics determine the corresponding free topologies. Such results are in a striking contrast with free balanced and free abelian topological groups cases (in standard varieties). Another...

Free spaces

Jian Song, E. Tymchatyn (2000)

Fundamenta Mathematicae

A space Y is called a free space if for each compactum X the set of maps with hereditarily indecomposable fibers is a dense G δ -subset of C(X,Y), the space of all continuous functions of X to Y. Levin proved that the interval I and the real line ℝ are free. Krasinkiewicz independently proved that each n-dimensional manifold M (n ≥ 1) is free and the product of any space with a free space is free. He also raised a number of questions about the extent of the class of free spaces. In this paper we will...

Fully closed maps and non-metrizable higher-dimensional Anderson-Choquet continua

Jerzy Krzempek (2010)

Colloquium Mathematicae

Fedorchuk's fully closed (continuous) maps and resolutions are applied in constructions of non-metrizable higher-dimensional analogues of Anderson, Choquet, and Cook's rigid continua. Certain theorems on dimension-lowering maps are proved for inductive dimensions and fully closed maps from spaces that need not be hereditarily normal, and some of the examples of continua we construct have non-coinciding dimensions.

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).

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