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Bing maps and finite-dimensional maps

Michael Levin (1996)

Fundamenta Mathematicae

Let X and Y be compacta and let f:X → Y be a k-dimensional map. In [5] Pasynkov stated that if Y is finite-dimensional then there exists a map g : X → 𝕀 k such that dim (f × g) = 0. The problem that we deal with in this note is whether or not the restriction on the dimension of Y in the Pasynkov theorem can be omitted. This problem is still open.  Without assuming that Y is finite-dimensional Sternfeld [6] proved that there exists a map g : X → 𝕀 k such that dim (f × g) = 1. We improve this result of Sternfeld showing...

Borsuk's quasi-equivalence is not transitive

Andrzej Kadlof, Nikola Koceić Bilan, Nikica Uglešić (2007)

Fundamenta Mathematicae

Borsuk's quasi-equivalence relation on the class of all compacta is considered. The open problem concerning transitivity of this relation is solved in the negative. Namely, three continua X, Y and Z lying in ℝ³ are constructed such that X is quasi-equivalent to Y and Y is quasi-equivalent to Z, while X is not quasi-equivalent to Z.

Borsuk-Sieklucki theorem in cohomological dimension theory

Margareta Boege, Jerzy Dydak, Rolando Jiménez, Akira Koyama, Evgeny V. Shchepin (2002)

Fundamenta Mathematicae

The Borsuk-Sieklucki theorem says that for every uncountable family X α α ∈ A of n-dimensional closed subsets of an n-dimensional ANR-compactum, there exist α ≠ β such that d i m ( X α ∩ X β ) = n . In this paper we show a cohomological version of that theorem: Theorem. Suppose a compactum X is c l c ℤ n + 1 , where n ≥ 1, and G is an Abelian group. Let X α α ∈ J be an uncountable family of closed subsets of X. If d i m G X = d i m G X α = n for all α ∈ J, then d i m G ( X α ∩ X β ) = n for some α ≠ β. For G being a countable principal ideal domain the above result was proved by Choi and Kozlowski...

C * -points vs P -points and P ♭ -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

In a Tychonoff space X , the point p ∈ X is called a C * -point if every real-valued continuous function on C ∖ { p } can be extended continuously to p . Every point in an extremally disconnected space is a C * -point. A classic example is the space 𝐖 * = ω 1 + 1 consisting of the countable ordinals together with ω 1 . The point ω 1 is known to be a C * -point as well as a P -point. We supply a characterization of C * -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space is a C * -point....

Cantor manifolds in the theory of transfinite dimension

Wojciech Olszewski (1994)

Fundamenta Mathematicae

For every countable non-limit ordinal α we construct an α-dimensional Cantor ind-manifold, i.e., a compact metrizable space Z α such that i n d Z α = α , and no closed subset L of Z α with ind L less than the predecessor of α is a partition in Z α . An α-dimensional Cantor Ind-manifold can be constructed similarly.

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