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𝒵 -distributive function lattices

Marcel Erné (2013)

Mathematica Bohemica

It is known that for a nonempty topological space X and a nonsingleton complete lattice Y endowed with the Scott topology, the partially ordered set [ X , Y ] of all continuous functions from X into Y is a continuous lattice if and only if both Y and the open set lattice 𝒪 X are continuous lattices. This result extends to certain classes of 𝒵 -distributive lattices, where 𝒵 is a subset system replacing the system 𝒟 of all directed subsets (for which the 𝒟 -distributive complete lattices are just the continuous...

A Basic Fixed Point Theorem

Lech Pasicki (2006)

Bulletin of the Polish Academy of Sciences. Mathematics

The paper contains a fixed point theorem for stable mappings in metric discus spaces (Theorem 10). A consequence is Theorem 11 which is a far-reaching extension of the fundamental result of Browder, Göhde and Kirk for non-expansive mappings.

A characterization of the meager ideal

Piotr Zakrzewski (2015)

Commentationes Mathematicae Universitatis Carolinae

We give a classical proof of the theorem stating that the σ -ideal of meager sets is the unique σ -ideal on a Polish group, generated by closed sets which is invariant under translations and ergodic.

A characterization of ω-limit sets for piecewise monotone maps of the interval

Andrew D. Barwell (2010)

Fundamenta Mathematicae

For a piecewise monotone map f on a compact interval I, we characterize the ω-limit sets that are bounded away from the post-critical points of f. If the pre-critical points of f are dense, for example when f is locally eventually onto, and Λ ⊂ I is closed, invariant and contains no post-critical point, then Λ is the ω-limit set of a point in I if and only if Λ is internally chain transitive in the sense of Hirsch, Smith and Zhao; the proof relies upon symbolic dynamics. By identifying points of...

A class of continua that are not attractors of any IFS

Marcin Kulczycki, Magdalena Nowak (2012)

Open Mathematics

This paper presents a sufficient condition for a continuum in ℝn to be embeddable in ℝn in such a way that its image is not an attractor of any iterated function system. An example of a continuum in ℝ2 that is not an attractor of any weak iterated function system is also given.

A class of spaces that admit no sensitive commutative group actions

Jiehua Mai, Enhui Shi (2012)

Fundamenta Mathematicae

We show that a metric space X admits no sensitive commutative group action if it satisfies the following two conditions: (1) X has property S, that is, for each ε > 0 there exists a cover of X which consists of finitely many connected sets with diameter less than ε; (2) X contains a free n-network, that is, there exists a nonempty open set W in X having no isolated point and n ∈ ℕ such that, for any nonempty open set U ⊂ W, there is a nonempty connected open set V ⊂ U such that the boundary X ( V ) ...

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