3,236 research outputs found
Adjoint entropy vs Topological entropy
Recently the adjoint algebraic entropy of endomorphisms of abelian groups was
introduced and studied. We generalize the notion of adjoint entropy to
continuous endomorphisms of topological abelian groups. Indeed, the adjoint
algebraic entropy is defined using the family of all finite-index subgroups,
while we take only the subfamily of all open finite-index subgroups to define
the topological adjoint entropy. This allows us to compare the (topological)
adjoint entropy with the known topological entropy of continuous endomorphisms
of compact abelian groups. In particular, the topological adjoint entropy and
the topological entropy coincide on continuous endomorphisms of totally
disconnected compact abelian groups. Moreover, we prove two Bridge Theorems
between the topological adjoint entropy and the algebraic entropy using
respectively the Pontryagin duality and the precompact duality.Comment: 18 page
Two Topological Uniqueness Theorems for Spaces of Real Numbers
A 1910 theorem of Brouwer characterizes the Cantor set as the unique totally
disconnected, compact metric space without isolated points. A 1920 theorem of
Sierpinski characterizes the rationals as the unique countable metric space
without isolated points. The purpose of this exposition is to give an
accessible overview of this celebrated pair of uniqueness results. It is
illuminating to treat the problems simultaneously because of commonalities in
their proofs. Some of the more counterintuitive implications of these results
are explored through examples. Additionally, near-examples are provided which
thwart various attempts to relax hypotheses.Comment: 11 page
Locally Minimal Topological Groups 2
We continue in this paper the study of locally minimal groups started in
\cite{LocMin}. The minimality criterion for dense subgroups of compact groups
is extended to local minimality. Using this criterion we characterize the
compact abelian groups containing dense countable locally minimal subgroups, as
well as those containing dense locally minimal subgroups of countable
free-rank. We also characterize the compact abelian groups whose torsion part
is dense and locally minimal. We call a topological group {\it almost
minimal} if it has a closed, minimal normal subgroup such that the quotient
group is uniformly free from small subgroups. The class of almost minimal
groups includes all locally compact groups, and is contained in the class of
locally minimal groups. On the other hand, we provide examples of countable
precompact metrizable locally minimal groups which are not almost minimal. Some
other significant properties of this new class are obtained
Schemes as functors on topological rings
Let be a scheme. In this text, we extend the known definitions of a
topology on the set of -rational points from topological fields,
local rings and ad\`ele rings to any ring with a topology. This definition
is functorial in both and , and it does not rely on any restriction on
like separability or finiteness conditions. We characterize properties of
, such as being a topological Hausdorff ring, a local ring or having
as an open subset for which inversion is continuous, in terms of
functorial properties of the topology of . Particular instances of this
general approach yield a new characterization of adelic topologies, and a
definition of topologies for higher local fields.Comment: 14 page
Compact Totally Disconnected Moufang Buildings
Let be a spherical building each of whose irreducible components is
infinite, has rank at least 2 and satisfies the Moufang condition. We show that
can be given the structure of a topological building that is compact
and totally disconnected precisely when is the building at infinity of
a locally finite affine building.Comment: To appear in Tohoku Math. Journa
Hereditarily h-complete groups
A topological group G is h-complete if every continuous homomorphic image of
G is (Raikov-)complete; we say that G is hereditarily h-complete if every
closed subgroup of G is h-complete. In this paper, we establish open-map
properties of hereditarily h-complete groups with respect to large classes of
groups, and prove a theorem on the (total) minimality of subdirectly
represented groups. Numerous applications are presented, among them: 1. Every
hereditarily h-complete group with quasi-invariant basis is the projective
limit of its metrizable quotients; 2. If every countable discrete hereditarily
h-complete group is finite, then every locally compact hereditarily h-complete
group that has small invariant neighborhoods is compact. In the sequel, several
open problems are formulated.Comment: 12 pages; few changes were made compared to the original submission
thanks to the suggestions of the refere
Decomposing locally compact groups into simple pieces
We present a contribution to the structure theory of locally compact groups.
The emphasis is on compactly generated locally compact groups which admit no
infinite discrete quotient. It is shown that such a group possesses a
characteristic cocompact subgroup which is either connected or admits a
non-compact non-discrete topologically simple quotient. We also provide a
description of characteristically simple groups and of groups all of whose
proper quotients are compact. We show that Noetherian locally compact groups
without infinite discrete quotient admit a subnormal series with all
subquotients compact, compactly generated Abelian, or compactly generated
topologically simple. Two appendices introduce results and examples around the
concept of quasi-product.Comment: Index added; minor change
- …