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W. W. Comfort - Continuous cross sections on abelian groups equipped with the Bohr topology
| W. W. COMFORT, Wesleyan University |
| Continuous cross sections on abelian groups equipped with the Bohr topology |
All groups here are Abelian. A closed subgroup H of a topological
group G is a
ccs-subgroup if there is a continuous cross
section from G/H to G--that is, a continuous function 
such that
(with
the natural homomorphism).
The symbol 
denotes G with its Bohr topology, i.e.,
the topology induced by
.
A topological group H is an absolute
ccs-group
[resp., an absolute retract ] if His a
ccs-subgroup [resp., is a retract] in every group of the form
containing H as a (necessarily closed) subgroup. One then
writes
[resp.,
].
Theorem 1. Every
ccs-subgroup H of a group of the form
is a retract of (and is homeomorphic to
); hence
.
Theorem 2.
[resp.,
] iff 
is a
ccs-subgroup of its
divisible hull
[resp., is a
retract of
].
Theorem 3. (a) Every cyclic group is in
.
(b) The classes
and
are closed
under finite products.
Theorem 4. Not every Abelian group is in
.
Question [van Douwen, 1990]. Is every Abelian group in
?
* Presented in Kingston by this co-author.
Next: Ilijas Farah - need Up: 3) Set Theoretic Topology / Previous: Krzysztof Ciesielski - Each
