Résume | The box topology on a product of topological spaces is given by declaring as basic open sets any products of open sets of the composants of the product.
It is a natural definition, but does not preserve many topological properties, notably it miserably fails to preserve compactness. However, a weakening of compactness, called paracompactness and introduced by Jean Dieudonné in 1944 for its nice behaviour in analysis, is sometimes preserved by box products. Investigating this for spaces obtained by the product of countably many factors or aleph_1 many factors with either full boxes or boxes of countable size, was a classical topic in set-theoretic topology of the 1980s or so, with important works by Mary Ellen Rudin, Kenneth Kunen, Eric van Douwen and others.
In our work in progress we are, rather, interested in an unexplored territory of products with many coordinates. In particular, we consider the following question:
Suppose that kappa is a cardinal such that for every lambda >= kappa, the box product {}^{<\kappa} 2^\lambda is paracompact.
Is kappa a large cardinal ?
(the notation means that the topology on 2^lambda is generated by boxes of size < kappa)
We present some partial results and the difficulties with the consideration of the case kappa singular. This is somewhat connected with the recent works on descriptive set theory of the space 2^kappa for kappa singular.
This is joint work with David Buhagiar, University of Malta. |