In this paper, we prove the following two statements: (1) There exists a discretely absolutely star-Lindelöf Tychonoff space having a regular-closed subspace which is not CCC-Lindelöf. (2) Every Hausdorff (regular, Tychonoff) linked-Lindelöf space can be represented in a Hausdorff (regular, Tychonoff) absolutely star-Lindelöf space as a closed Gδ subspace.

We give easy proofs that a) the Continuum Hypothesis implies that if the product of X with every Lindelöf space is Lindelöf, then X is a D-space, and b) Borel’s Conjecture implies every Rothberger space is Hurewicz.

In this paper, we prove the following statements: (1) There exist a Tychonoff space X and a subspace Y of X such that Y is strongly star-Lindelöf in X and e(Y, X) is arbitrarily large, but X is not star-Lindelöf. (2) There exist a Tychonoff space X and a subspace Y of X such that Y is star-Lindelöf in X, but Y is not strongly star-Lindelöf in X.

We answer questions of Bennett, Lutzer, and Matveev by showing that any monotonically compact LOTS is metrizable, and any first-countable Lindelöf GO-space is monotoncically Lindelöf. We also show that any compact monotonically Lindelöf space is first-countable, and is metrizable if scattered, and that separable monotonically compact spaces are metrizable.

We obtain from the consistency of the existence of a measurable cardinal the consistency of “small” upper bounds on the cardinality of a large class of Lindelöf spaces whose singletons are Gδ sets. Call a topological space in which each singleton is a Gδ set a points Gδ space. A.V. Arhangel’skii proved that any points Gδ Lindelöf space must have cardinality less than the least measurable cardin...

There is a locally compact Hausdorff space of weight אω which is linearly Lindelöf and not Lindelöf.

We obtain several results and examples concerning the general question “When must a space with a small diagonal have a Gδ-diagonal?”. In particular, we show (1) every compact metrizably fibered space with a small diagonal is metrizable; (2) there are consistent examples of regular Lindelöf (even hereditarily Lindelöf) spaces with a small diagonal but no Gδ-diagonal; (3) every first-countable he...

Arhangel’skii proved that if a first countable Hausdorff space is Lindelöf, then its cardinality is at most 20 . Such a clean upper bound for Lindelöf spaces in the larger class of spaces whose points are Gδ has been more elusive. In this paper we continue the agenda started in [50], of considering the cardinality problem for spaces satisfying stronger versions of the Lindelöf property. Infinit...

Some new results about linearly Lindelöf spaces are given here. It is proved that if X is a space of countable spread and X = Y ∪ Z, where Y and Z are meta-Lindelöf spaces, then X is linearly Lindelöf. Moreover, we give a positive answer to a problem raised by A.V. Arhangel’skii and R.Z. Buzyakova.

No convenient internal characterization of spaces that are productively Lindelöf is known. Perhaps the best general result known is Alster’s internal characterization, under the Continuum Hypothesis, of productively Lindelöf spaces which have a basis of cardinality at most א1. It turns out that topological spaces having Alster’s property are also productively weakly Lindelöf. The weakly Lindelö...