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The Baire property in remainders of topological groups and other results

Aleksander V. Arhangel'skii (2009)

Commentationes Mathematicae Universitatis Carolinae

It is established that a remainder of a non-locally compact topological group G has the Baire property if and only if the space G is not Čech-complete. We also show that if G is a non-locally compact topological group of countable tightness, then either G is submetrizable, or G is the Čech-Stone remainder of an arbitrary remainder Y of G . It follows that if G and H are non-submetrizable topological groups of countable tightness such that some remainders of G and H are homeomorphic, then the spaces...

The Banach contraction mapping principle and cohomology

Ludvík Janoš (2000)

Commentationes Mathematicae Universitatis Carolinae

By a dynamical system ( X , T ) we mean the action of the semigroup ( ℤ + , + ) on a metrizable topological space X induced by a continuous selfmap T : X → X . Let M ( X ) denote the set of all compatible metrics on the space X . Our main objective is to show that a selfmap T of a compact space X is a Banach contraction relative to some d 1 ∈ M ( X ) if and only if there exists some d 2 ∈ M ( X ) which, regarded as a 1 -cocycle of the system ( X , T ) × ( X , T ) , is a coboundary.

The equivariant universality and couniversality of the Cantor cube

Michael G. Megrelishvili, Tzvi Scarr (2001)

Fundamenta Mathematicae

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...

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