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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.

Centers of a dendroid

Jo Heath, Van C. Nall (2006)

Fundamenta Mathematicae

A bottleneck in a dendroid is a continuum that intersects every arc connecting two non-empty open sets. Piotr Minc proved that every dendroid contains a point, which we call a center, contained in arbitrarily small bottlenecks. We study the effect that the set of centers in a dendroid has on its structure. We find that the set of centers is arc connected, that a dendroid with only one center has uncountably many arc components in the complement of the center, and that, in this case, every open set...

Chaotic continua of (continuum-wise) expansive homeomorphisms and chaos in the sense of Li and Yorke

Hisao Kato (1994)

Fundamenta Mathematicae

A homeomorphism f : X → X of a compactum X is expansive (resp. continuum-wise expansive) if there is c > 0 such that if x, y ∈ X and x ≠ y (resp. if A is a nondegenerate subcontinuum of X), then there is n ∈ ℤ such that d ( f n ( x ) , f n ( y ) ) > c (resp. d i a m f n ( A ) > c ). We prove the following theorem: If f is a continuum-wise expansive homeomorphism of a compactum X and the covering dimension of X is positive (dim X > 0), then there exists a σ-chaotic continuum Z = Z(σ) of f (σ = s or σ = u), i.e. Z is a nondegenerate subcontinuum...

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