Countable Subcover Research Articles

$\sigma$-locales in Formal Topology

A $\sigma$-frame is a poset with countable joins and finite meets in which binary meets distribute over countable joins. The aim of this paper is to show that $\sigma$-frames, actually $\sigma$-locales, can be seen as a branch of Formal Topology, that is, intuitionistic and predicative point-free topology. Every $\sigma$-frame $L$ is the lattice of Lindel\"of elements (those for which each of their covers admits a countable subcover) of a formal topology of a specific kind which, in its turn, is a presentation of the free frame over $L$. We then give a constructive characterization of the smallest (strongly) dense $\sigma$-sublocale of a given $\sigma$-locale, thus providing a "$\sigma$-version" of a Boolean locale. Our development depends on the axiom of countable choice.

Some remarks on Star-Menger spaces using box products

This article is a continuation of study of star-Menger selection properties in line of (Kocinac, 2009, 2015), where we mainly use covers consisting of G? sets with certain additional condition. It is observed that star-Mengerness is equivalent to the fact that every such type of cover of a space has a countable subcover. We improve this result by considering ?subcovers of cardinality less than b? instead of ?countable subcovers?, which is our primary observation. We also show that it is possible to produce non normal spaces using box products and dense star-Menger subspaces.

Some versions of second countability of metric spaces in ZF and their role to compactness

In the realm of metric spaces we show in ZF that: (i) A metric space is compact if and only if it is countably compact and for every $\varepsilon > 0$, every cover by open balls of radius $\varepsilon $ has a countable subcover. (ii) Every second countable metric space has a countable base consisting of open balls if and only if the axiom of countable choice restricted to subsets of $\mathbb{R}$ holds true. (iii) A countably compact metric space is separable if and only if it is second countable.

A first countable linearly Lindelöf not Lindelöf topological space

A topological space X is called linearly Lindelöf if every increasing open cover of X has a countable subcover. It is well known that every Lindelöf space is linearly Lindelöf. The converse implication holds only in particular cases, such as X being countably paracompact or if n w ( X ) < ℵ ω . Arhangelʼskii and Buzyakova proved that the cardinality of a first countable linearly Lindelöf space does not exceed 2 ℵ 0 . Consequently, a first countable linearly Lindelöf space is Lindelöf if ℵ ω > 2 ℵ 0 . They asked whether every linearly Lindelöf first countable space is Lindelöf in ZFC. This question is supported by the fact that all known linearly Lindelöf not Lindelöf spaces are of character at least ℵ ω . We answer this question in the negative by constructing a counterexample from M A + ℵ ω < 2 ℵ 0 . A modification of Alsterʼs Michael space that is first countable is presented.

Mappings and some decompositions of continuity on nearly Lindelöf spaces

A topological space (X, π) is said to be nearly Lindelof if every regularly open cover of (X, π) has a countable subcover. In this paper we study the effect of mappings and some decompositions of continuity on nearly Lindelof spaces. The main result is that a δ-continuous image of a nearly Lindelof space is nearly Lindelof.

An example concerning the property of a space being Lindelöf in another

A space X is said to be Lindelöf in a space Z if every open cover of Z has a countable subcover of X. Ranchin asks if there must always be a Lindelöf Y with X ⊂ Y ⊂ Z. We answer this in the negative.

The metrisability of precompact sets

A uniform space is trans-separable if every uniform cover has a countable subcover. We show that a uniform space is trans-separable if it contains a suitable family of precompact sets. Applying this result to locally convex spaces, we are able to deduce that the precompact subsets of a wide class of spaces are metrisable. The proof of our main Theorem is based on a cardinality argument, and is reminiscent of the classical Bolzano-Weierstrass Theorem.

Lindelöf models of the reals: Solution to a problem of Sikorski

We show that it is consistent with ZFC that there is a modelM of ZF + DC such that the integers ofM areω1-like, the reals ofM have cardinalityω2, and the unit interval [0, 1]M is Lindelof (i.e. every open cover has a countable subcover). This answers an old question of Sikorski.

Properties of weak 𝜃̄-refinable spaces

A space X X is called weak θ ¯ \overline \theta -refinable if every open cover of X X has a refinement ⋃ i = 1 ∞ G i \bigcup \nolimits _{i = 1}^\infty {{\mathcal {G}_i}} satisfying (1) G i = { G α : α ϵ A i } {\mathcal {G}_i} = \{ {G_\alpha }:\alpha \epsilon {A_i}\} is an open collection for each i i , (2) each x ϵ X x\epsilon X has finite positive order with respect to some G i {\mathcal {G}_i} , (3) the open cover { G i = ⋃ [ G α : α ϵ A i ] } i = 1 ∞ \{ {G_i} = \bigcup {[{G_\alpha }:\alpha \epsilon } {A_i}]\} _{i = 1}^\infty is point finite. In this paper the author shows that the above property lies between the properties of θ \theta -refinable and weak θ \theta -refinable. The main result is the fact that if X X is countably metacompact and satisfies property ( δ ) (\delta ) , every weak θ ¯ \overline \theta -cover of X X has a countable subcover. Results concerning paracompactness, metacompactness and the star-finite property are also obtained.

A generalization of a theorem of Aquaro

In this paper we introduce the concept of a δθ-cover to generalize Aquaro's Theorem that every point countable open cover of a topological space such that every discrete closed family of sets is countable has a countable subcover. A δθ-cover of a space X is defined to be a family of open sets where each Vn covers X and for x є X there exists n such that Vn is of countable order at x. We replace point countable open cover by a δθ-cover in Aquaro's Theorem and also generalize the result of Worrell and Wicke that a θ-refinable countably compact space is compact and Jones′ result that ℵ1-compact Moore space is Lindelöf which was used to prove his classic result that a normal separable Moore space is metrizable, using the continuum hypothesis.