Open Problem Seminar
Norm and pointwise topologies need not be binormal.
Logarithmic capacity is not subadditive – a fine topology approach
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 of strongly self-homeomorphic dendrite not pointwise self-homeomorphic
Such spaces in which a homeomorphic image of the whole space can be found in every open set are called . W.J. Charatonik and A. Dilks asked if any strongly self-homeomorphic dendrite is pointwise self-homeomorphic. We give a negative answer in Example 2.1.
An example for mappings related to confluence
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.
An example related to strongly pointwise self-homeomorphic dendrites
Such spaces in which a homeomorphic image of the whole space can be found in every open set are called self-homeomorphic. W.J. Charatonik and A. Dilks posed a problem related to strongly pointwise self-homeomorphic dendrites. We solve this problem negatively in Example 2.1.
Normal spaces and the Lusin-Menchoff property
We study the relation between the Lusin-Menchoff property and the -“semiseparation” property of a fine topology in normal spaces. Three examples of normal topological spaces having the -“semiseparation” property without the Lusin-Menchoff property are given. A positive result is obtained in the countable compact space.
Waraszkiewicz spirals revisited
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).
An example for mappings related to semi-confluence.
On the sense preserving mappings in the Helm topology in the plane
∗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.
Chain of dendrites without open supremum.
Chain of dendrites without retract infimum.
Chain of dendrites without open infimum.
Chain of dendrites openly unbounded from below.
On piecewise confluent mappings.
Irreducible mappings and HU-terminal continua.
Irreducible mapping and the lightness of open mappings.
An arcwise connected continuum without nonboundary proper arcwise connected subcontinua.
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