The equivariant universality and couniversality of the Cantor cube

Michael G. Megrelishvili; Tzvi Scarr

Fundamenta Mathematicae (2001)

  • Volume: 167, Issue: 3, page 269-275
  • ISSN: 0016-2736

Abstract

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Let ⟨G,X,α⟩ be a G-space, where G is a non-Archimedean (having a local base at the identity consisting of open subgroups) and second countable topological group, and X is a zero-dimensional compact metrizable space. Let ⟨ H ( 0 , 1 ℵ ₀ ) , 0 , 1 ℵ ₀ , τ ⟩ be the natural (evaluation) action of the full group of autohomeomorphisms of the Cantor cube. Then (1) there exists a topological group embedding φ : G ↪ H ( 0 , 1 ℵ ₀ ) ; (2) there exists an embedding ψ : X ↪ 0 , 1 ℵ ₀ , equivariant with respect to φ, such that ψ(X) is an equivariant retract of 0 , 1 ℵ ₀ with respect to φ and ψ.

How to cite

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Michael G. Megrelishvili, and Tzvi Scarr. "The equivariant universality and couniversality of the Cantor cube." Fundamenta Mathematicae 167.3 (2001): 269-275. <http://eudml.org/doc/281728>.

@article{MichaelG2001,
abstract = {Let ⟨G,X,α⟩ be a G-space, where G is a non-Archimedean (having a local base at the identity consisting of open subgroups) and second countable topological group, and X is a zero-dimensional compact metrizable space. Let $⟨H(\{0,1\}^\{ℵ₀\}),\{0,1\}^\{ℵ₀\},τ⟩$ be the natural (evaluation) action of the full group of autohomeomorphisms of the Cantor cube. Then (1) there exists a topological group embedding $φ: G ↪ H(\{0,1\}^\{ℵ₀\})$; (2) there exists an embedding $ψ: X ↪ \{0,1\}^\{ℵ₀\}$, equivariant with respect to φ, such that ψ(X) is an equivariant retract of $\{0,1\}^\{ℵ₀\}$ with respect to φ and ψ.},
author = {Michael G. Megrelishvili, Tzvi Scarr},
journal = {Fundamenta Mathematicae},
keywords = {Cantor cube; -compactification; universal space; non-Archimedean group},
language = {eng},
number = {3},
pages = {269-275},
title = {The equivariant universality and couniversality of the Cantor cube},
url = {http://eudml.org/doc/281728},
volume = {167},
year = {2001},
}

TY - JOUR
AU - Michael G. Megrelishvili
AU - Tzvi Scarr
TI - The equivariant universality and couniversality of the Cantor cube
JO - Fundamenta Mathematicae
PY - 2001
VL - 167
IS - 3
SP - 269
EP - 275
AB - Let ⟨G,X,α⟩ be a G-space, where G is a non-Archimedean (having a local base at the identity consisting of open subgroups) and second countable topological group, and X is a zero-dimensional compact metrizable space. Let $⟨H({0,1}^{ℵ₀}),{0,1}^{ℵ₀},τ⟩$ be the natural (evaluation) action of the full group of autohomeomorphisms of the Cantor cube. Then (1) there exists a topological group embedding $φ: G ↪ H({0,1}^{ℵ₀})$; (2) there exists an embedding $ψ: X ↪ {0,1}^{ℵ₀}$, equivariant with respect to φ, such that ψ(X) is an equivariant retract of ${0,1}^{ℵ₀}$ with respect to φ and ψ.
LA - eng
KW - Cantor cube; -compactification; universal space; non-Archimedean group
UR - http://eudml.org/doc/281728
ER -

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