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<sub>\alpha</sub>)] subset or equal U<sub>\beta</sub> when \alpha <= \beta. Given such a sequence \Sigma = <U<sub>\alpha</sub> | \alpha < \omega<sub>1</sub>>, 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

1. containing a copy of \omega₁; or
2. which is \omega₁-compact?

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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: April 5, 1998