Generating compact Hausdorff spaces using finite T₀-spaces
by
Richard G. Wilson
Universidad Autónoma Metropolitana
Coauthors: Ralph D. Kopperman

An old result of Flachsmeyer [F], recently resurrected in bitopological language in [KW1], states that every compact T₂-space is the Hausdorff reflection of the limit of an inverse spectrum of finite T₀-spaces. In this talk we will study the relation between properties of the compact Hausdorff space X_H, those of the inverse limit X, and those of the finite T₀-spaces of the spectrum. For example, if X is a chainable continuum, then all of the finite spaces may be chosen to be COTS (see [KKW]) while if X is a metric continuum of dimension n, then all of the spaces in the spectrum may be chosen to have Alexandroff dimension n (see [KW2]). Further, a compact T₂-space is the Hausdorff reflection of an inverse limit of finite normal T₀-spaces if and only if it is zero dimensional. By way of contrast however we have shown in [KW3] that some properties of the inverse limit may be determined by properties of the bonding maps: Let us say that a map f:S --> T is normalizing if inverse images of disjoint closed sets in T can be separated by open sets in S and chaining if f(cl(x)) is a specialization chain for each x in S. Then the following are equivalent for a spectral space X:

a) X is normal,

b) X is the inverse limit of finite T₀-spaces with normalizing bonding maps,

c) the Hausdorff reflection X_H of X is (homeomorphic to) the subspace of specialization minimal elements of X,

d) Each point of X contains a unique closed point in its closure.

Furthermore, the inverse limit X is a completely normal spectral space if and only if it is the inverse limit of finite T₀-spaces with bonding maps which are chaining. However, every compact T₂-space is the Hausdorff reflection of the inverse limit of a spectrum of finite T₀-spaces with bonding maps which are chaining.

Bibliography

[F] J. Flachsmeyer, Zur Spektralentwicklung topologischer Räume, Math. Annalen 144 (1961), 253-274.

[KKW] J. Kennedy, R. D. Kopperman and R.G. Wilson, The chainable continua are the spaces approximated by COTS, to appear in Applied General Topology, 2001.

[KW1] R. D. Kopperman and R. G. Wilson, Finite approximation of compact Hausdorff spaces, Topology Proceedings 22 (1997), 175-200.

[KW2] R. D. Kopperman and R. G. Wilson, The approximation of finite dimensional compacta by T₀-spaces, submitted for publication.

[KW3] R. D. Kopperman and R. G. Wilson, On the role of finite normal spaces and maps in the genesis of compact Hausdorff spaces, submitted for publication.

Date received: June 25, 2001

Copyright © 2001 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 # cagx-26.