Continuum many Fréchet types of hereditarily strongly infinite-dimensional Cantor manifolds

Chatyrko, Vitalij A.; Pol, Elżbieta

doi:10.1090/s0002-9939-99-05089-3

Cited by 6 publications

(1 citation statement)

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“…There are previous studies on the structure of continuous isomorphism types (Fréchet dimension types) of various kinds of infinite-dimensional compacta; for example, strongly infinite-dimensional Cantor manifolds (see [7, 8]). For instance, by combining the Baire category theorem and the result by Chatyrko-Pol [8], one can show that there are continuum many first-level Borel isomorphism types of strongly infinite-dimensional Cantor manifolds. However, there is an enormous gap between first and second level and, hence, such an argument never tells us anything about second-level Borel isomorphism types.…”

Section: Preliminariesmentioning

confidence: 99%

Point Degree Spectra of Represented Spaces

Kihara

Pauly

2022

Forum of Mathematics, Sigma

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We introduce the point degree spectrum of a represented space as a substructure of the Medvedev degrees, which integrates the notion of Turing degrees, enumeration degrees, continuous degrees and so on. The notion of point degree spectrum creates a connection among various areas of mathematics, including computability theory, descriptive set theory, infinite-dimensional topology and Banach space theory. Through this new connection, for instance, we construct a family of continuum many infinite-dimensional Cantor manifolds with property C whose Borel structures at an arbitrary finite rank are mutually nonisomorphic. This resolves a long-standing question by Jayne and strengthens various theorems in infinite-dimensional topology such as Pol’s solution to Alexandrov’s old problem.

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Section: Preliminariesmentioning

confidence: 99%

Point Degree Spectra of Represented Spaces

Kihara

Pauly

2022

Forum of Mathematics, Sigma

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Collections of higher-dimensional hereditarily indecomposable continua, with incomparable Fréchet types

Pol¹

2005

Topology and its Applications

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We construct a family {Y s : s ∈ S} of cardinality 2 ℵ 0 of hereditarily indecomposable continua which are: (a) n-dimensional Cantor manifolds, for any given natural number n, or (b) hereditarily strongly infinite-dimensional Cantor manifolds, or else (c) countable-dimensional continua of every given transfinite inductive dimension, small or large, such that if h : Y s → Y s is an embedding then s = s and h is the identity.

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