How large can a hereditarily separable or hereditarily Lindelöf space be?

The main result of this paper is that ifV satisfies GCH andω ₁<λ<μ are arbitrary regular cardinals, then in some cardinal preserving forcing extensionW ofV we have λ^<λ=λ=2 ^N 0,μ=2 ^λ and there are a hereditarily separableX ?2 ^λ with |X|= \(2^{\aleph _0 } \) =μ and a hereditarily LindelöfY ?2 ^μ withw(Y)= \(2^{2^\aleph 0} \) =μ. So far similar results have only been obtained under the assumption of CH.     

Resolvability of Lindelöf spaces

We prove the ω-resolvability of hereditarily finally compact spaces and the resolvability of Lindelöf spaces whose dispersion character is uncountable.     

The Lindelöf principle in ℂ n

Let D be a bounded domain in ? ⁿ . A holomorphic function f: D → ? is called normal function if f satisfies a Lipschitz condition with respect to the Kobayashi metric on D and the spherical metric on the Riemann sphere ??. We formulate and prove a few Lindelöf principles in the function theory of several complex variables.     

The Lindelöf Number of Functional Spaces on Monolithic Compacta

Let X be a compactum, τ be an infinite cardinal, and t(X) ≤ τ. In this case, l(C_p(X)) ≤ 2^τ. If X is τ-monolitliic, then l(C_p(X)) ≤ τ⁺. In addition, if X is zero-dimensional and there are no τ⁺-Aronszajn trees, then l(C_p(X)) ≤ τ.     

Some consequences of the Lindelöf conjecture

Suppose that the Lindelöf conjecture is valid in the following quantitative form: $$|\zeta (\frac{1}{2} + it)| \leqslant c_0 |t|^{\varepsilon (|t|)} $$ , where ε(t) is a monotone decreasing function, $\varepsilon (2t) \geqslant \tfrac{1}{2}\varepsilon (t),\varepsilon (t) \geqslant \tfrac{1}{{\sqrt {log t} }}$ . Then it is proved that for |t|≥T₀ the disk $\{ s:|s - \tfrac{1}{2} - it| \leqslant v\} $ contains at most 20v log |t| zeros of ζ(s) if $\tfrac{1}{2} \geqslant v \geqslant \sqrt {\varepsilon (t)} $ . There exists an absolute constant A such that for |t|≥T₁ the disk $\{ s:|s - \tfrac{1}{2} - it| \leqslant A\varepsilon ^{\tfrac{1}{3}} (t)\} $ contains at least one zero of ζ(s). Bibliography: 2 titles.     

A generalization of the Lindelöf theorem

We present a generalization of the Lindelöf theorem on conditions that should be imposed on the coefficients of the Taylor series of an entire transcendental function ? in order that the relation $ln M_f (r) - \tau r^\rho , r \to \infty , M_f (r) = \max \left\{ {\left| {f(r)} \right|:|z| = r} \right\}$ , be satisfied.     

Some generalizations of Lindelöf and paracompact spaces

In this paper notions ofm-Lindelöf, meta-m-Lindelöf, para-m-Lindelöf andm-closure preserving property are defined, wherem is any infinite cardinal. The main results are the following:

1. A topological space ism-Lindelöf if and only if it is meta-m-Lindelöf and it ism-Lindelöf in the sense of complete accumulation point.
2. A regular topological space is paracompact if and only if it is para-m-Lindelöf and it hasm-closure preserving property for somem.

     

Ernst Lindelöf in memoriam

Ohne Zusammenfassung     

Remarks on Completely Regular Lindelöf Reflection of Locales

We prove, under the axiom of countable choice, that the l-ring structure of C(L) of real-valued continuous functions on a locale L determines the completely regular Lindelöf reflection lL of L up to isomorphism. Mathematics Subject Classifications (2000) 54C30, 46E25.Project supported by NSF of China (Grant No. 10271056 and Grant No. 10331011).     

A broader view of the almost Lindelöf property

By first finding necessary and sufficient conditions for the realcompact coreflection, νL, and the regular Lindelöf coreflection, λL, of a completely regular frame L to be isomorphic, we define a frame L to be almost Lindelöf if it is Lindelöf or λL → L is a one-point extension. This agrees with the condition “νL is Lindelöf and L is realcompact or νL is a one-point extension”, which would be a frame version of what are called almost Lindelöf spaces. Thus, the condition “νX is Lindelöf”, which is added in the definition of almost Lindelöf spaces, serves only to compensate for the lack of the regular Lindelöf reflection in Top, and can be dispensed with by concentrating on the frame \({\mathfrak {O}X}\) instead of the space X.     

On nonlinear analogues of the Phragmen–Lindelöf theorem

For solutions of quasilinear elliptic inequalities containing lower-order derivatives, we obtain estimates of the growth that take into account the geometry of the domain. Corollaries of these results are nonlinear analogues of the Phragmen–Lindelöf theorem.     

On the product of relatively weakly almost Lindelöf subsets

The purpose of this note is to show that there exist two Tychonoff spaces X, Y, a subset A of X and a subset B of Y such that A is weakly almost Lindelöf in X and B is weakly almost Lindelöf in Y, but A × B is not weakly almost Lindelöf in X × Y.     

An Intrinsic Characterization of Monomorphisms in Regular Lindelöf Locales

We characterize monomorphisms in rLLoc, the category of regular Lindelöf locales. Though somewhat complicated, the characterization is intrinsic in the sense that it refers only to the properties of the morphism itself, rather than to properties of some lifting of it to a distant category.     

A note on weakly Lindelöf frames

The notion of σ^?-properness of a subset of a frame is introduced. Using this notion, we give necessary and su?cient conditions for a frame to be weakly Lindelöf. We show that a frame is weakly Lindelöf if and only if its semiregularization is weakly Lindelöf. For a completely regular frame L, we introduce a condition equivalent to weak realcompactness based on maximal ideals of the cozero part of L. This enables us to show that every weakly realcompact almost P -frame is realcompact. A new characterization of weakly Lindelöf frames in terms of neighbourhood strongly divisible ideals of ?? is provided. The closed ideals of ?? equipped with the uniform topology are applied to describe weakly Lindelöf frames.     

The Frölicher spectral sequence can be arbitrarily non-degenerate

The Frölicher spectral sequence of a compact complex manifold X measures the difference between Dolbeault cohomology and de Rham cohomology. If X is Kähler then the spectral sequence collapses at the E ₁term and no example with d _n  ≠  0 for n > 3 has been described in the literature.We construct for n ≥  2 nilmanifolds with left-invariant complex structure X _n such that the n-th differential d _n does not vanish. This answers a question mentioned in the book of Griffiths and Harris.     

On countable unions of nonmeager sets in hereditarily Lindelöf spaces

It is well known that any Vitali set on the real line ? does not possess the Baire property. The same is valid for finite unions of Vitali sets. What can be said about infinite unions of Vitali sets? Let S be a Vitali set, S _r be the image of S under the translation of ? by a rational number r and F = {S _r : r is rational}. We prove that for each non-empty proper subfamily F′ of F the union ∪F′ does not possess the Baire property. We say that a subset A of ? possesses Vitali property if there exist a non-empty open set O and a meager set M such that A ? O \ M. Then we characterize those non-empty proper subfamilies F′ of F which unions ∪F′ possess the Vitali property.     

Compactifications in which the collection of multiple points is Lindelöf semi-stratifiable

We show that every compactification of a topological space in which the collection of multiple points is Lindelöf semi-stratifiable is a z-compactification. In particular such a compactification is a Wallman compactification.     

Some remarks on generalizations of countably compact spaces and Lindelöf spaces

We prove some propositions and present some examples concerning the properties between countably compact and pseudocompact and the properties between Lindelöf and pseudo-Lindelöf.