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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...

On Surjective Bing Maps

Hisao Kato, Eiichi Matsuhashi (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

In [7], M. Levin proved that the set of all Bing maps of a compact metric space to the unit interval is a dense G δ -subset of the space of all maps. In [6], J. Krasinkiewicz independently proved that the set of all Bing maps of a compact metric space to an n-dimensional manifold (n ≥ 1) is a dense G δ -subset of the space of maps. In [9], J. Song and E. D. Tymchatyn, solving some problems of J. Krasinkiewicz ([6]), proved that the set of all Bing maps of a compact metric space to a nondegenerate connected...

On the almost monotone convergence of sequences of continuous functions

Zbigniew Grande (2011)

Open Mathematics

A sequence (f n)n of functions f n: X → ℝ almost decreases (increases) to a function f: X → ℝ if it pointwise converges to f and for each point x ∈ X there is a positive integer n(x) such that f n+1(x) ≤ f n (x) (f n+1(x) ≥ f n(x)) for n ≥ n(x). In this article I investigate this convergence in some families of continuous functions.

On topological and algebraic structure of extremally disconnected semitopological groups

Aleksander V. Arhangel'skii (2000)

Commentationes Mathematicae Universitatis Carolinae

Starting with a very simple proof of Frol’ık’s theorem on homeomorphisms of extremally disconnected spaces, we show how this theorem implies a well known result of Malychin: that every extremally disconnected topological group contains an open and closed subgroup, consisting of elements of order 2 . We also apply Frol’ık’s theorem to obtain some further theorems on the structure of extremally disconnected topological groups and of semitopological groups with continuous inverse. In particular, every...

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