We present a forcing construction of a Hausdorff zero-dimensional Lindelöf space X whose square X is again Lindelöf but its cube X has a closed discrete subspace of size c, hence the Lindelöf degree L(X) = c. In our model the Continuum Hypothesis holds true. After that we give a description of a forcing notion to get a space X such that L(X) = א0 for all positive integers n, but L(X0 ) = c = א2.

We present some new methods for constructing a Michael space, a regular Lindelöf space which has a non-Lindelöf product with the space of irrationals. The central result is a combinatorial statement about the irrationals which is a necessary and sufficient condition for the existence of a certain class of Michael spaces. We also show that there are Michael spaces assuming d = cov(M) and that it...

There is a locally compact Hausdorff space which is linearly Lindelöf and not Lindelöf. This answers a question of Arhangel’skii and Buzyakova.

We define ω-directedness, investigate various properties to determine whether they have this property or not, and use our results to obtain easier proofs of theorems due to Laurence and Alster concerning the existence of a Michael space, i.e. a Lindelöf space whose product with the irrationals is not Lindelöf.

If B is a compact space and B \ {pt} is Lindelöf then Bκ \ { − → pt} is star-Linedlöf for every κ. If B \ {pt} is compact then Bκ \ { − → pt} is discretely star-Lindelöf. In particular, this gives new examples of Tychonoff discretely star-Lindelöf spaces with unlimited extent.

Let X be the Bennett-Lutzer’s space and Y be the space obtained from X by shrinking the set of all rational numbers to a point. In his book, G.Gao claimed that the space Y is compact. In this paper, we prove that Y is neither countably compact nor Lindelöf, which shows that G.Gao’s claim is not true. Moreover, we prove that Y is strongly paracompact. As an application of this result, we obtain ...

We study conditions on a topological space that guarantee that its product with every Lindelöf space is Lindelöf. The main tool is a condition discovered by K. Alster and we call spaces satisfying his condition Alster spaces. We also study some variations on scattered spaces that are relevant for this question.

A cardinal λ is called ω-inaccessible if for all μ < λ we have μ < λ. We show that for every ω-inaccessible cardinal λ there is a CCC (hence cardinality and cofinality preserving) forcing that adds a hereditarily Lindelöf regular space of density λ. This extends an analogous earlier result of ours that only worked for regular λ. In [1] we have shown that for any cardinal λ a natural CCC forcing...