180 research outputs found
An example of a non-Borel locally-connected finite-dimensional topological group
Answering a question posed by S.Maillot in MathOverFlow, for every
we construct a locally connected subgroup of dimension , which is not locally compact.Comment: 2 page
On closed embeddings of free topological algebras
Let be a complete quasivariety of completely regular universal
topological algebras of continuous signature (which means that
is closed under taking subalgebras, Cartesian products, and
includes all completely regular topological -algebras algebraically
isomorphic to members of ). For a topological space by
we denote the free universal -algebra over in the class
. Using some extension properties of the Hartman-Mycielski
construction we prove that for a closed subspace of a metrizable (more
generally, stratifiable) space the induced homomorphism
between the respective free universal algebras is a closed topological
embedding. This generalizes one result of V.Uspenskii concerning embeddings of
free topological groups.Comment: 3 page
The topology of systems of hyperspaces determined by dimension functions
Given a non-degenerate Peano continuum , a dimension function
defined on the family of compact subsets of ,
and a subset , we recognize the topological structure
of the system (2^X,\D_{\le\gamma}(X))_{\alpha\in\Gamma}, where is the
hyperspace of non-empty compact subsets of and is the
subspace of , consisting of non-empty compact subsets with
.Comment: 12 page
The coarse classification of countable abelian groups
We prove that two countable locally finite-by-abelian groups G,H endowed with
proper left-invariant metrics are coarsely equivalent if and only if their
asymptotic dimensions coincide and the groups are either both
finitely-generated or both are infinitely generated. On the other hand, we show
that each countable group G that coarsely embeds into a countable abelian group
is locally nilpotent-by-finite. Moreover, the group G is locally
abelian-by-finite if and only if G is undistorted in the sense that G can be
written as the union of countably many finitely generated subgroups G_n such
that each G_n is undistorted in G_{n+1} (which means that the identity
inclusion from G_n to G_{n+1} is a quasi-isometric embedding with respect to
word metrics).Comment: 25 pages. Longer version with new results about FCC groups, locally
finite-by-abelian groups, locally nilpotent-by-finite groups
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