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On a compactification of the homeomorphism group of the pseudo-arc

Kazuhiro Kawamura (1991)

Colloquium Mathematicae

A continuum means a compact connected metric space. For a continuum X, H(X) denotes the space of all homeomorphisms of X with the compact-open topology. It is well known that H(X) is a completely metrizable, separable topological group. J. Kennedy [8] considered a compactification of H(X) and studied its properties when X has various types of homogeneity. In this paper we are concerned with the compactification G P of the homeomorphism group of the pseudo-arc P, which is obtained by the method of...

On hit-and-miss hyperspace topologies

Gerald Beer, Robert K. Tamaki (1993)

Commentationes Mathematicae Universitatis Carolinae

The Vietoris topology and Fell topologies on the closed subsets of a Hausdorff uniform space are prototypes for hit-and-miss hyperspace topologies, having as a subbase all closed sets that hit a variable open set, plus all closed sets that miss (= fail to intersect) a variable closed set belonging to a prescribed family Δ of closed sets. In the case of the Fell topology, where Δ consists of the compact sets, a closed set A misses a member B of Δ if and only if A is far from B in a uniform sense....

On indecomposability and composants of chaotic continua

Hisao Kato (1996)

Fundamenta Mathematicae

A homeomorphism f:X → X of a compactum X with metric d is expansive if there is c > 0 such that if x,y ∈ X and x ≠ y, then there is an integer n ∈ ℤ such that d ( f n ( x ) , f n ( y ) ) > c . A homeomorphism f: X → X is continuum-wise expansive if there is c > 0 such that if A is a nondegenerate subcontinuum of X, then there is an integer n ∈ ℤ such that d i a m i f n ( A ) > c . Clearly, every expansive homeomorphism is continuum-wise expansive, but the converse assertion is not true. In [6], we defined the notion of chaotic continua of homeomorphisms...

On Mazurkiewicz sets

Marta N. Charatonik, Włodzimierz J. Charatonik (2000)

Commentationes Mathematicae Universitatis Carolinae

A Mazurkiewicz set M is a subset of a plane with the property that each straight line intersects M in exactly two points. We modify the original construction to obtain a Mazurkiewicz set which does not contain vertices of an equilateral triangle or a square. This answers some questions by L.D. Loveland and S.M. Loveland. We also use similar methods to construct a bounded noncompact, nonconnected generalized Mazurkiewicz set.

On multifunctions with closed graphs

D. Holý (2001)

Mathematica Bohemica

The set of points of upper semicontinuity of multi-valued mappings with a closed graph is studied. A topology on the space of multi-valued mappings with a closed graph is introduced.

On reflexive closed set lattices

Zhongqiang Yang, Dong Sheng Zhao (2010)

Commentationes Mathematicae Universitatis Carolinae

For a topological space X , let S ( X ) denote the set of all closed subsets in X , and let C ( X ) denote the set of all continuous maps f : X → X . A family 𝒜 ⊆ S ( X ) is called reflexive if there exists 𝒞 ⊆ C ( X ) such that 𝒜 = { A ∈ S ( X ) : f ( A ) ⊆ A for every f ∈ 𝒞 } . Every reflexive family of closed sets in space X forms a sub complete lattice of the lattice of all closed sets in X . In this paper, we continue to study the reflexive families of closed sets in various types of topological spaces. More necessary and sufficient conditions for certain families of closed...

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