Quasi-Developable Manifolds
by
A. M. Mohamad
The University of Auckland

There is a quasi-developable 2-manifold with a G_\delta diagonal, which is not developable. Consistently, the example can be made to be countably metacompact. As it is natural to ask what is required to get from quasi developability to developability, Peter Nyikos in Questions 10 and 11 asked:

(1) Is there a quasi-developable manifold with a G_\delta-diagonal that is not developable?
(2) Is every countably metacompact, quasi-developable manifold with a G_\delta-diagonal Moore space?

In this paper we answer both of these questions in the negative by constructing appropriate manifolds, and the reader is referred there for the background to Nyikos' problem.

Recall that a space is an n-manifold if it is Hausdorff, connected and has an open cover by subspaces homeomorphic to Rⁿ. One can easily check that a space is developable if and only if it is quasi-developable and perfect (every closed subset is a G_\delta). A space is countably metacompact if every countable open cover has a point-finite open refinement. Perfect spaces are countably metacompact.

Perhaps the most striking feature of our examples is that they are highly geometric (in contrast to most pathological manifolds), with the topology defined `all in one go'. This simplifies the task of calculating stars of points in open families.

Both examples are derived from locally compact, locally countable `sub-real' (in other words, obtained by refining the usual topology on the real line) spaces, which have the desired topological properties. The first such sub-real space is due to Gruenhage (Example 2.17 ). But the second sub-real space (which, consistently, is locally compact, quasi-developable, countably metacompact, submetrizable, but not perfect) is new.

In the first part of this paper we give the constructions of the manifolds from the sub-real spaces. The second part is devoted to the creation of the second sub-real space. The space is based upon an example of Balogh and Burke , which has the requisite properties, except for the fact that it is not sub-real. It seems likely that an example of Shelah can be modified to give a consistent example with stronger properties: it is sub-real, locally compact, quasi-developable, normal, countably paracompact, but not perfect.

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Z.T. Balogh & D.K. Burke, A total ladder system space by ccc forcing, Topology and its Applications 44 (1992) 37-44.

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G. Gruenhage, Generalized metric spaces, Handbook of Set-Theoretic Topology, North-Holland, (1984), 423 - 501.

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P. Nyikos, The Theory of Nonmetrizable Manifolds, Handbook of Set-Theoretic Topology, North-Holland, (1984), 633 - 684.

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P. Nyikos, Mary Ellen Rudin's Contributions to the Theory of Nonmetrizable Manifolds, The work of Mary Ellen Rudin, Ann. New York Acad. Sci., 705 (1993), 92 - 113.

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S. Shelah, A consistent counterexample in the theory of collectionwise Hausdorff spaces, Israel Journal of Mathematics 65 (1989) 219-224.

Date received: May 31, 1999

Copyright © 1999 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 # cacc-26.