Set-theoretic constructions of nonshrinking open covers (Q1063893)

The authors give two interesting examples of topological spaces. (1) Let \(\kappa\) be an infinite regular cardinal. Then \(\diamond^{++}\) implies the existence of a Hausdorff, strongly zero-dimensional, collectionwise normal, \(\kappa\)-ultraparacompact, P-space with the following property: Every increasing open cover has a clopen shrinking, but there is an open cover which has no closed shrinking. (2) Let \(\kappa\) be an uncountable regular cardinal. Then \(\Delta\) implies the existence of a Hausdorff, collectionwise normal, countably ultraparacompact P-space, which has a strictly increasing open cover having no shrinking, each member of which is the union of at most \(\kappa\) closed sets.