Alignments and the theory of locally compact spaces
by
Peter J. Nyikos
University of South Carolina

The concept of an alignment is a way of formalizing the fact that surprisingly general kinds of spaces are, informally speaking, narrow.

An alignment of a space X is a family <X_\alpha : \alpha < \theta> of open subspaces of X whose union is X, such that [`(X_\alpha)] is a proper subset of X_\beta whenever \alpha < \beta. The ordinal \theta is called the length of the alignment, while the width at \alpha of the alignment equals the Lindelöf degree of X_\alpha \ \cup _{\xi < \alpha} X_\xi, and the width of the alignment is the supremum of the widths at all ordinals < \theta. Here are some results in this area:

Theorem 1 Every locally compact, locally connected space satisfying wD hereditarily has a continuous alignment of width <= \omega₁ such that the boundary of X_\alpha is closed discrete for all ordinals \alpha of uncountable cofinality.

Under strong set-theoretic hypotheses, and the strengthening of ßatisfying wD hereditarily" to "hereditarily normal and hereditarily collectionwise Hausdorff", one can replace \omega₁ in this theorem with \omega and omit all conditions on \alpha. Corollaries are that such spaces are hereditarily countably paracompact and hereditarily collectionwise normal, as well as the speaker's recent result that all hereditarily normal, hereditarily cwH manifolds of dimension greater than 1 are metrizable under the set-theoretic hypotheses employed. This is also a corollary of:

Theorem 2 If it is consistent that there is a supercompact cardinal, it is consistent that every locally compact, hereditarily normal, heditarily cwH space X of Lindelöf degree \aleph₁ has a continuous alignment of width \omega and length \omega₁ such that the boundary of each X_\alpha is countable.

By adding \omega₁-compactness to the hypotheses of Theorem 2, one can get the boundary of each X_\alpha to be discrete as well, leading to the conclusion that these spaces are collectionwise normal and countably paracompact under the given set-theoretic hypotheses.

Date received: March 2, 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 # caca-54.