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Compactifications, Hartman functions and (weak) almost periodicity

100%

Maresch G., Winkler R.

Instytut Matematyczny Polskiej Akademii Nauk

We investigate Hartman functions on a topological group G. Recall that (ι,C) is a group compactification of G if C is a compact group, ι: G → C is a continuous group homomorphism and ι(G) ⊆ C is dense. A bounded function f: G → ℂ is a Hartman function if there exists a group compactification (ι,C) and F: C → ℂ such that f = F∘ι and F is Riemann integrable, i.e. the set of discontinuities of F is a null set with respect to the Haar measure. In particular, we determine how large a compactification for a given group G and a Hartman function f: G → ℂ must be to admit a Riemann integrable representation of f. The connection to (weakly) almost periodic functions is investigated. In order to give a systematic presentation which is self-contained to a reasonable extent, we include several separate sections on the underlying concepts such as finitely additive measures on Boolean set algebras, means on algebras of functions, integration on compact spaces, compactifications of groups and semigroups, the Riemann integral on abstract spaces, invariance of measures and means, continuous extensions of transformations and operations to compactifications, etc.

Fundamental groups of one-dimensional spaces

76%

Dorfer G., Thuswaldner J., Winkler R.

Fundamenta Mathematicae

2013

tom 223

nr 2

137-169

Let X be a metrizable one-dimensional continuum. We describe the fundamental group of X as a subgroup of its Čech homotopy group. In particular, the elements of the Čech homotopy group are represented by sequences of words. Among these sequences the elements of the fundamental group are characterized by a simple stabilization condition. This description of the fundamental group is used to give a new algebro-combinatorial proof of a result due to Eda on continuity properties of homomorphisms from the fundamental group of the Hawaiian earring to that of X.

Ograniczanie wyników

1 Fundamenta Mathematicae

2 Winkler R.

1 Dorfer G.

1 Maresch G.

1 Thuswaldner J.

1 2013

1 2009

Wyniki wyszukiwania

Compactifications, Hartman functions and (weak) almost periodicity

Fundamental groups of one-dimensional spaces