274 research outputs found
Compactifications of topological groups
Every topological group has some natural compactifications which can be a
useful tool of studying . We discuss the following constructions: (1) the
greatest ambit is the compactification corresponding to the algebra of
all right uniformly continuous bounded functions on ; (2) the Roelcke
compactification corresponds to the algebra of functions which are both
left and right uniformly continuous; (3) the weakly almost periodic
compactification is the envelopping compact semitopological semigroup of
(`semitopological' means that the multiplication is separately continuous).
The universal minimal compact -space is characterized by the
following properties: (1) has no proper closed -invariant subsets; (2)
for every compact -space there exists a -map . A group is
extremely amenable, or has the fixed point on compacta property, if is a
singleton. We discuss some results and questions by V. Pestov and E. Glasner on
extremely amenable groups. The Roelcke compactifications were used by M.
Megrelishvili to prove that can be a singleton. They can be used to
prove that certain groups are minimal. A topological group is minimal if it
does not admit a strictly coarser Hausdorff group topology.Comment: 17 page
On subgroups of minimal topological groups
A topological group is minimal if it does not admit a strictly coarser
Hausdorff group topology. The Roelcke uniformity (or lower uniformity) on a
topological group is the greatest lower bound of the left and right
uniformities. A group is Roelcke-precompact if it is precompact with respect to
the Roelcke uniformity. Many naturally arising non-Abelian topological groups
are Roelcke-precompact and hence have a natural compactification. We use such
compactifications to prove that some groups of isometries are minimal. In
particular, if U_1 is the Urysohn universal metric space of diameter 1, the
group Iso(U_1) of all self-isometries of U_1 is Roelcke-precompact,
topologically simple and minimal. We also show that every topological group is
a subgroup of a minimal topologically simple Roelcke-precompact group of the
form Iso(M), where M is an appropriate non-separable version of the Urysohn
space.Comment: To appear in Topology and its Applications. 39 page
The Roelcke compactification of groups of homeomorphisms
Let X be a zero-dimensional compact space such that all non-empty clopen
subsets of X are homeomorphic to each other, and let H(X) be the group of all
self-homeomorphisms of X with the compact-open topology. We prove that the
Roelcke compactification of H(X) can be identified with the semigroup of all
closed relations on X whose domain and range are equal to X. We use this to
prove that the group H(X) is topologically simple and minimal, in the sense
that it does not admit a strictly coarser Hausdorff group topology.Comment: 9 pages. To appear in Topology App
A note on a question of R. Pol concerning light maps
Let f:X -> Y be an onto map between compact spaces such that all
point-inverses of f are zero-dimensional. Let A be the set of all functions u:X
-> I=[0,1] such that is zero-dimensional for all y in Y.
Do almost all maps u:X -> I, in the sense of Baire category, belong to A? H.
Toru\'nczyk proved that the answer is yes if Y is countable-dimensional. We
extend this result to the case when Y has property C.Comment: 4 pages. Topology Appl. (to appear
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