The complexity of the set of squares in the homeomorphism group of the circle
The set of squares in the group of autohomeomorphisms of the circle is complete analytic, and hence analytic but not Borel.
The complexity of -discretely decomposable families in uniform spaces
The concept of boundedness and the Bohr compactification of a MAP Abelian group
Let G be a maximally almost periodic (MAP) Abelian group and let ℬ be a boundedness on G in the sense of Vilenkin. We study the relations between ℬ and the Bohr topology of G for some well known groups with boundedness (G,ℬ). As an application, we prove that the Bohr topology of a topological group which is topologically isomorphic to the direct product of a locally convex space and an -group, contains “many” discrete C-embedded subsets which are C*-embedded in their Bohr compactification. This...
The concept of differential equation in topological spaces and generalized mechanics.
The congruences on S(X) which commute with V.
The Conley index for decompositions of isolated invariant sets
The control agglomerations of an internet network with delay feedback.
The convergence of mean value iteration for a family of maps.
The covering property for σ-ideals of compact, sets
The covering property for σ-ideals of compact sets is an abstract version of the classical perfect set theorem for analytic sets. We will study its consequences using as a paradigm the σ-ideal of countable closed subsets of .
The dimension of analytic sets
The dual group of a dense subgroup
Throughout this abstract, is a topological Abelian group and is the space of continuous homomorphisms from into the circle group in the compact-open topology. A dense subgroup of is said to determine if the (necessarily continuous) surjective isomorphism given by is a homeomorphism, and is determined if each dense subgroup of determines . The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable group is...
The dual space of precompact groups
For any topological group the dual object is defined as the set of equivalence classes of irreducible unitary representations of equipped with the Fell topology. If is compact, is discrete. In an earlier paper we proved that is discrete for every metrizable precompact group, i.e. a dense subgroup of a compact metrizable group. We generalize this result to the case when is an almost metrizable precompact group.
The Dugundji extension property can fail in ωµ -metrizable spaces
We show that there exist -metrizable spaces which do not have the Dugundji extension property ( with the countable box topology is such a space). This answers a question posed by the second author in 1972, and shows that certain results of van Douwen and Borges are false.
The dynamics of piecewise monotonic maps under small perturbations
The entropy on -quantum spaces
The equicontinuous structure relation of a unicoherent point-transitive flow
The equivalence between the convergences of Mann and Ishikawa iteration methods with errors for demicontinuous -strongly accretive operators in uniformly smooth Banach spaces.
The equivariant universality and couniversality of the Cantor cube
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 be the natural (evaluation) action of the full group of autohomeomorphisms of the Cantor cube. Then (1) there exists a topological group embedding ; (2) there exists an embedding , equivariant with respect to φ, such that ψ(X) is an equivariant retract of with respect to φ...
The existence of initially -compact group topologies on free Abelian groups is independent of ZFC
It was known that free Abelian groups do not admit a Hausdorff compact group topology. Tkachenko showed in 1990 that, under CH, a free Abelian group of size admits a Hausdorff countably compact group topology. We show that no Hausdorff group topology on a free Abelian group makes its -th power countably compact. In particular, a free Abelian group does not admit a Hausdorff -compact nor a sequentially compact group topology. Under CH, we show that a free Abelian group does not admit a Hausdorff...
The extending topologies.