When does the 0-dimensional Reflection Preserve Products? by Andrej Bauer

Topology Atlas Document # qaaa-19 | Production Editor: Harriet Lord

Question
When does the 0-dimensional Reflection Preserve Products?
Andrej Bauer

A Question about Zero-dimensional Spaces

We say that a topological space is zero-dimensional when the collection of its clopen (closed and open) subsets forms a basis for its topology.

For a topological space X, let z(X) be its zero-dimensional reflection, i.e., the same underlying set X but with the topology generated by the collection of the clopen subsets of X.

Question 1: Does z preserve products, i.e., is z(X ×Y) homeomorphic to z(X) ×z(Y)?

The question can be stated equivalently as follows.

Question 1': Is every clopen subset of X ×Y a union of the sets of the form U ×V, where U 'subset or equal' X is clopen and V 'subset or equal' Y is clopen?

We know that the answer to this is negative. Raouchan Bouziakova constructed a couple of counter-examples. Alexander Arhangel'skii observed that the answer is affirmative if X is compact.

For the purposes of the original motivation, it would be useful to know the answer to the following question.

Question 2: Does Question 1 have an affirmative answer if X and Y are homogeneous, and are topological quotients of homogenous, countably based, T₀-spaces?

Background and Motivation

The motivation for this question comes from theoretical computer science. Consider the category of equilogical spaces [1], originally defined by Dana Scott, which is a cartesian closed extension of the category of T₀-spaces. Briefly, the objects are pairs (X, ~ ) where X is a T₀-spaces and ~ is an equivalence relation on X. A morphism is (an equivalence class of) an equivalence preserving continuous maps, i.e., a map f\colon X --> Y such that a ~ _X b implies f(a) ~ _Y f(b). See the reference for details.

Every T₀-space X can be viewed as an equilogical space equipped with the identity equivalence relation. Because the category of equilogical spaces is cartesian closed, we can iterate exponentials of the space of the real numbers with Euclidean topology, R:

R, R^R,R^{R^R}, ...

The object R^R is isomorphic to the space of continuous real functions with the compact-open topology. In the same category, we can construct the real numbers object (see a standard reference on topos theory, e.g. [2]. It turns out to be the equilogical space R_I = (C, ~ ), where C is the space of rapidly converging Cauchy sequences of rational numbers,

C = { f \colon N --> Q | 'for all'k. (|f(k+1) - f(k)| < 2^-k) } ,

equipped with the topology inherited from the Baire space Q^N. The equivalence relation ~ is the usual coincidence of Cauchy sequences,

f ~ g <===> 'for all'k. (|f(k) - g(k)| < 2^-k+2) .

Aside: the subscript I stands for ``intensional reals''. We can form iterated exponentials of R_I as well:

R_I, R_I^R_I,R_I^{R_I^R_I}, ...

The question arises, whether the two hierarchies of higher types over R and R_I coincide. At level zero, we see that R is the topological quotient of C by ~ , so the answer is affirmative: there is a natural bijective correspondence between real numbers and equivalence classes of Cauchy sequences (this is rather obvious).

Martín Escardó, Alex Simpson, and I checked that this coincidence extends to levels one and two, i.e., there is a natural bijective correspondence between R^R and R_I^R_I, and also between R^{R^R} and R_I^{R_I^R_I}. We do not know the answer for higher types.

Alex Simpson and I observed the following: if the zero-dimensional reflection preserves products, then the hierarchies of Euclidean reals and intensional reals coincide.

For other background material on this topic see [3].

A. Bauer, L. Birkedal, D. Scott, ``Equilogical Spaces'', 1998, Preprint submitted to Elsevier, available at
http://www.cs.cmu.edu/Groups/LTC/
S. Mac Lane, I. Moerdijk, ``Sheaves in Geometry and Logic. A First Introduction to Topos Theory'', Springer-Verlag, New York, 1992
A. Simpson, M. Menni, ``Topological and Limit-space Subcategories of Countably-based Equilogical Spaces'', extended journal version of MFPS XV, submitted, available at
http://www.dcs.ed.ac.uk/home/als/Research/

Received by the editors: April 10, 2000
Revised: May 4, 2000