Finite Approximation of Compact Hausdorff Spaces
by
R.G. Wilson
Universidad Autónoma Metropolitana, México
Coauthors: R. D. Kopperman (Universidad Autónoma Metropolitana, México)

More than thirty years ago, Flachsmeyer [F] suggested one approach to the study of compact Hausdorff spaces, namely, to approximate the compact space by means of finite T₀-spaces. His work went largely unnoticed by topologists - possibly because compact Hausdorff spaces are in some sense the best in the field, whereas finite T₀-spaces are T₁ only if they are discrete.

Today however, with computers widespread the use of ``large finite" objects is commonplace. Whenever you look at a computer screen, you are essentially looking at a finite topological space. Indeed, although the computer screen looks like a product of two intervals, much use is made of the fact that the pixels can be modelled as elements of the product of two finite sets. In [KKM] this situation is studied from a topological point of view. An interval in the Khalimsky line is an example of a finite connected ordered topological space (COTS); such spaces have frequently been used as finite models of intervals in the reals. However, our goal here is not simply to study finite spaces, but to use them to approximate other (infinite) topological spaces.

In order to develop a theory of finite approximations, we shall restrict our attention to those asymmetric spaces (X, \tau), in which a second topology, \tau^D (called a dual of \tau) is defined, such that the bitopological space (X, \tau, \tau^D) exhibits some `better' bitopological separation property than the original topological space. For example, in a finite space, the collection of closed sets forms such a topology, which we will call \tau^G. We will see that if (X, \tau) is a finite T₀-space then the bitopological space (X, \tau, \tau^G) mimics properties of a compact Hausdorff space in an obvious way. All terminology is taken from [K] or [KKM].

Let us say that a T₂-space X, is finitely approximable if it is the Hausdorff reflection of an inverse limit of finite T₀-spaces and quotient maps. Among other reults, we will show that:

Theorem 1. If X is a skew compact space, then its T₂ -reflection X_H is its quotient by the smallest symmetrically closed equivalence relation containing <= _\tau(ie., the intersection of all symmetrically closed equivalence relations containing <= _\tau). Furthermore, X_H is the T_i-reflection of X for 1 <= i <= 4.

Theorem 2. Given any compact Hausdorff space, X, there is an inverse limit Y of finite T₀-spaces and quotient maps, together with a de Groot quotient p:Y --> X. The map p is a retract of Y onto an ^SG-dense subspace, through which each map from Y into a T₂-space factors. Thus a T₂ space is finitely approximable if and only if it is compact.

Theorem 3. Suppose (X, \tau) is a Hausdorff. Then the following are equivalent:

(a) (X, \tau) is a continuum.

(b) (X, \tau) has an approximation by finite connected spaces.

(c) (X, \tau) has a connected spectral compactification.

We will also show that approximators of compact Hausdorff spaces also arise in a familiar way as Stone spaces of Boolean algebras.

REFERENCES

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

[K] R. D. Kopperman, `Asymmetry and duality in topology', Topology Appl. 66 (1995), 1-39.

[KKM] E. Khalimsky, R. D. Kopperman and P. R. Meyer, `Computer graphics and connected topologies on finite ordered sets', Topology Appl. 36 (1990), 1-17.

Date received: July 4, 1997

Copyright © 1997 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 Mathematical Conference Abstracts. Document # caao-86.