Atlas: Constructing type-I manifolds with given $\Upsilon$-trees by Sina Greenwood

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The Second Galway Topology Colloquium at Oxford
September 2-5, 1998
The University of Oxford
Oxford, UK

Organizers
Chris Good, Paul Gartside, Peter Collins, Steven Fisher

Constructing type-I manifolds with given \Upsilon-trees
by
Sina Greenwood
University of Auckland

Nyikos [] defines a Type-I manifold to be a manifold M which can be written in the form \cup _{\alpha in \omega₁}U_\alpha, where each U_\alpha is open and Lindelöf, [`(U_\alpha)] subset or equal U_\beta when \alpha <= \beta. Given such a sequence \Sigma = <U_\alpha | \alpha < \omega₁>, he defines the tree of nonmetrizable component boundaries associated with \Sigma, denoted \Upsilon(\Sigma), to be the collection of all sets of the form \partialC such that C is a nonmetrizable componenent of M-[`(U_\alpha)] for some \alpha, with the following order: if A, B in \Upsilon(\Sigma), then A <= B if and only if B is a subset of the component whose boundary is A.

We will discuss the problem of, given a particular tree T, constructing a manifold M having T as its \Upsilon-tree. In particular, we will address the following questions: given a well-pruned \omega₁-tree T, can we always find such a manifold M

containing a copy of \omega₁; or

which is \omega₁-compact?

References

[]: P. Nyikos, The Theory of Nonmetrizable Manifolds, in: Handbook of Set-Theoretic Topology, K. Kunen and J. E. Vaughan, eds, North-Holland, Amsterdam (1984), 633-684

Date received: August 6, 1998