Kuratowski-Ulam pairs of topological spaces
by
Tomasz Natkaniec
Department of Mathematics, Gdansk University, Poland
Coauthors: David Fremlin (Math. Dep., University of Essex, England), Ireneusz Reclaw (Dep. Math., Gdansk University, Poland)

This talk is based on the joint paper [FNR].

A pair (X, Y) of topological spaces is called Kuratowski-Ulam pair if the Kuratowski-Ulam Theorem holds in X×Y, i.e.,

If E in M(X×Y), then { x in X : E_x not in M(Y)} in M(X)

Y is called universally Kuratowski-Ulam space, if (X, Y) is a Kuratowski-Ulam pair for every space X. Obviously every meager in itself space is uKU. Moreover, it is known that every space with a countable \pi-basis is uKU. We prove the following:

• Every dyadic space (in fact, any continuous image of any product of separable metrizable spaces) is uKU space, so there are uKU Baire spaces which do not have countable \pi-bases.
• Every Baire uKU space is ccc.
• There exist Baire, ccc, non uKU spaces.

A space Y is called really Kuratowski-Ulam space, if (R, Y) is a KU pair. Obviously, uKU subset rKU. We prove that every rKU space is c-cc. Thus under CH every rKU space is ccc, but it is consistent that there exists a Baire rKU space which is non-ccc. Thus it is consistent that uKU =/= rKU. We do not know whether this inequality can be proved in ZFC.

References

[FRN] D. Fremlin, T. Natkaniec, I. Recaw, Universally Kuratowski- Ulam spaces, Fund. Math., to appear.

Date received: June 27, 2000