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On the sense preserving mappings in the Helm topology in the plane

Pyrih, Pavel — 1999

Serdica Mathematical Journal

∗Research supported by the grant No. GAUK 186/96 of Charles University. We introduce the helm topology in the plane. We show that (assuming the helm local injectivity and the Euclidean continuity) each mapping which is oriented at all points of a helm domain U is oriented at U.

Logarithmic capacity is not subadditive – a fine topology approach

Pavel Pyrih — 1992

Commentationes Mathematicae Universitatis Carolinae

In Landkof’s monograph [8, p. 213] it is asserted that logarithmic capacity is strongly subadditive, and therefore that it is a Choquet capacity. An example demonstrating that logarithmic capacity is not even subadditive can be found e.gi̇n [6, Example 7.20], see also [3, p. 803]. In this paper we will show this fact with the help of the fine topology in potential theory.

An example for mappings related to confluence

Pavel Pyrih — 1999

Archivum Mathematicum

Confluence of a mapping between topological spaces can be defined by several ways. J.J. Charatonik asked if two definitions of the confluence using the components and quasi-components are equivalent for surjective mappings with compact point inverses. We give the negative answer to this question in Example 2.1.

Normal spaces and the Lusin-Menchoff property

Pavel Pyrih — 1997

Mathematica Bohemica

We study the relation between the Lusin-Menchoff property and the F σ -“semiseparation” property of a fine topology in normal spaces. Three examples of normal topological spaces having the F σ -“semiseparation” property without the Lusin-Menchoff property are given. A positive result is obtained in the countable compact space.

Waraszkiewicz spirals revisited

Pavel PyrihBenjamin Vejnar — 2012

Fundamenta Mathematicae

We study compactifications of a ray with remainder a simple closed curve. We give necessary and sufficient conditions for the existence of a bijective (resp. surjective) mapping between two such continua. Using those conditions we present a simple proof of the existence of an uncountable family of plane continua no one of which can be continuously mapped onto any other (the first such family, so called Waraszkiewicz's spirals, was created by Z. Waraszkiewicz in the 1930's).

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