Assuming Jensen’s principle ♦, there is a compact Hausdorff space X which is hereditarily Lindelöf, hereditarily separable, and connected, such that no closed subspace of X is both perfect and totally disconnected. The Proper Forcing Axiom implies that there is no such space. The ♦ example also fails to satisfy the CSWP (the complex version of the Stone-Weierstrass Theorem). This space cannot c...

Alternative growth and decay estimates, reminiscent of the classical Phragmén-Lindelöf principle, are derived for a linearised thermoelastic body whose plane cross-sections increase unboundedly with respect to a given direction. The proof uses a modified Poincaré inequality to construct a differential inequality for a weighted linear combination of the cross-sectional mechanical and thermal ene...

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.

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...

We discuss relationships in Lindelöf spaces among the properties “Menger”, “Hurewicz”, “Alster”, “productive”, and “D”. This note is a continuation of [13]. The question of what additional assumptions ensure that the product of two Lindelöf spaces is Lindelöf is natural and well-studied. See e.g., [28], [30], [2], [3], [4], [5], [6], [32], [33]. D-spaces were introduced in [20]. Definition. A s...

ζ(0 + it) = O(|t| 1 2 + ), for each > 0. Apply the Phragmén–Lindelöf Theorem to ζ(s) in a small interval containing 1. Clearly, we can make the linear function arising from the Phragmén–Lindelöf Theorem to pass arbitrarily close to 0 at σ = 1. Thus, as |t| → ∞, ζ(1 + it) = O(|t| ), for each > 0. To apply the Phragmén–Lindelöf Theorem to ζ(s) in the strip 0 ≤ σ ≤ 1, we set k(σ) = aσ + b. Thus, k...

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.

For a topological space X let K(X) be the set of all compact subsets of X. The purpose of this paper is to characterize Lindelöf Čech-complete spaces X by means of the sets K(X). Similar characterizations also hold for Lindelöf locally compact X, as well as for countably K-determined spaces X. Our results extend a classical result of J. Christensen.

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 ...