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Some results on CO spaces
by
Robert Bonnet
Laboratoire de Mathématiques, CNRS-UMR-5127, Université de Savoie, Chambéry, France
Coauthors: Matatyahu Rubin
A topological space X is called a CO space, if every closed subset of X is homeomorphic to some clopen subset of X. For instance, every successor ordinal with its order topology is a compact CO space.
In Bonnet and Rubin [BR],
we give a complete classification of CO spaces which are continuous images
of compact ordered spaces.
We prove first that a CO space
which is a continuous image of a compact interval space
must be scattered
and then we describe such a space as a finite disjoint union
of compact spaces of the following form:
(i) the one-point compactification Xℵ1
of a discrete space of cardinality ℵ1;
(ii) the interval space l+ 1 + m*, where l
and m are regular cardinals
(m* denotes the ordinal m with its converse ordering); and
(iii) a "sufficiently big" compact ordinal space.
So, such spaces are near to be compact ordinals.
Two main questions arise.
(1) Is there a non-scattered compact Hausdorff CO space? It is even not known whether it is consistent with ZFC that such a space exists. S. Shelah, in [S], gives some information about compact CO spaces.
(2) Does it follow from ZFC that there is a compact Hausdorff CO space which does not belong to the class defined above? S. Shelah showed in [BS] that, under ♢w1, there is a thin tall CO space. Note that this space is not of the above form.
[BR] R. Bonnet and M. Rubin:
A classification of CO spaces which are continuous images of compact ordered spaces,
Topology and its Applications, 155 (2008), pp: 375-411.
[BS] R. Bonnet and S. Shelah :
On HCO spaces. An uncountable compact T2 space, different from
ℵ1+1, is a thin tall compact space which is homeomorphic to each of its uncountable closed subspaces,
Israel J. Math. 84 (1993), no.3, pp. 289-332.
[S] S. Shelah :
Factor = quotient, uncountable Boolean algebras,
number of endomorphism and width, Math. Japonica 37 (1992),
pp. 385-400.
References
Date received: June 2, 2008
Copyright © 2008 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caxg-03.