We present a forcing construction of a Hausdorff zero-dimensional Lindelöf space whose square is again Lindelöf but its cube has a closed discrete subspace of size , hence the Lindelöf degree . In our model the Continuum Hypothesis holds true. After that we give a description of a forcing notion to get a space such that for all positive integers , but .
We show that given infinite sets and a function which is onto and -to-one for some , the preimage of any ultrafilter of under extends to an ultrafilter. We prove that the latter result is, in some sense, the best possible by constructing a permutation model with a set of atoms and a finite-to-one onto function such that for each free ultrafilter of its preimage under does not extend to an ultrafilter. In addition, we show that in there exists an ultrafilter compact pseudometric...
We study in ZF and in the class of spaces the web of implications/ non-implications between the notions of pseudocompactness, light compactness, countable compactness and some of their ZFC equivalents.