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The classical density topology is the topology generated by the lower density operator connected with a density point of a set. One of the possible generalizations of the concept of a density point is replacing the Lebesgue measure by the outer Lebesgue measure. It turns out that the analogous family associated with such generalized density points is not a topology. In this case, one can prove that this family is a strong generalized topology. In the paper some properties of the strong generalized topology connected with density points with respect to the outer Lebesgue measure will be presented. Moreover, among others, some characterizations of the families of meager sets and compact sets in this space will be given.

Graphings are special bounded-degree graphs on probability spaces, representing limits of graph sequences that are convergent in a local or local-global sense. We describe a procedure for turning the underlying space into a compact metric space, where the edge set is closed and nearby points have nearby graph neighborhoods., Comment: 10 pages; improved version of an auxiliary result from arXiv:1709.03179v1

We introduce the notion of a cellular-compact space and prove that cellular compactness is a nice property that implies cellular Lindelofness. The class of cellular-compact spaces is preserved by continuous images and finite unions, as well as by regular closed subsets and extensions. It is established that cellular-compact spaces must be pseudocompact but not necessarily countably compact. We prove that first countable cellular-compact regular spaces are countably compact and their cardinality does not exceed $${2^{\omega}}$$ . We also show that a collectionwise normal Frechet–Urysohn cellular-compact space need not be compact and there exist Frechet–Urysohn cellular-compact spaces that do not have a dense countably compact subspace.

A $$\Delta_1$$ -completion of a poset is a completion for which, simultaneously, every element is reachable as a join of meets and a meet of joins from the original poset. We focus our attention on $$\Delta_1$$ -completions that can be obtained from polarities $$\langle\mathcal{F},\mathcal{I},\mathcal{R}\rangle$$ where $$\mathcal{F}$$ is a collection of upsets containing the principal upsets, $$\mathcal{I}$$ is a collection of downsets containing the principal downsets of the original poset, and $$R\subseteq\mathcal{F}\times\mathcal{I}$$ is the relation of nonempty intersection. These $$\Delta_1$$ -completions are called $$\langle\mathcal{F},\mathcal{I}\rangle$$ -completions, and they satisfy a compactness property. In this paper, we show that if a pair $$\langle\mathcal{F},\mathcal{I}\rangle$$ satisfies a separating condition (similar to the Prime Filter Theorem for distributive lattices), then the $$\langle\mathcal{F},\mathcal{I}\rangle$$ -completion of the original poset is a completely distributive algebraic lattice. Given a poset P and an algebraic closure system $$\mathcal{F}$$ of upsets of P satisfying a distributivity condition, we show how to choose a collection of downsets $$\mathcal{I}$$ of P such that the $$\langle\mathcal{F},\mathcal{I}\rangle$$ -completion of P is a completely distributive algebraic lattice. Then, we study the extensions of additional operations on posets to their corresponding $$\langle\mathcal{F},\mathcal{I}\rangle$$ -completions. Finally, we use the previous results to obtain adequate $$\langle\mathcal{F},\mathcal{I}\rangle$$ -completions for the classes of Tarski algebras and Hilbert algebras, and for the classes of algebras that are canonically associated (in the sense of abstract algebraic logic) with some propositional logics.

Motivated by the density condition in the sense of Heinrich for Frechet spaces and by some results of Schluchtermann and Wheeler for Banach spaces, we characterize in terms of certain weakly compact resolutions those Frechet spaces enjoying the property that each bounded subset of its Mackey* dual is metrizable. We also characterize those Kothe echelon Frechet spaces $${\lambda _{p}(A)}$$ as well as those Frechet spaces Ck (X) of real-valued continuous functions equipped with the compact-open topology that enjoy this property.

We solve a long standing question due to Arhangel’skii by constructing a compact space which has a $${G_\delta}$$ cover with no continuum-sized ( $${G_\delta}$$ )-dense subcollection. We also prove that in a countably compact weakly Lindelof normal space of countable tightness, every $${G_\delta}$$ cover has a $${\mathfrak{c}}$$ -sized subcollection with a $${G_\delta}$$ -dense union and that in a Lindelof space with a base of multiplicity continuum, every $${G_\delta}$$ cover has a continuum sized subcover. We finally apply our results to obtain a bound on the cardinality of homogeneous spaces which refines De la Vega’s celebrated theorem on the cardinality of homogeneous compacta of countable tightness.

We study c-compactness in two cases. First, we obtain some subclasses of locally compact groups where compactness and c-compactness coincide, and besides, a result due to Dikranjan and Uspenskij is generalized. We introduce c-compactness and h-completeness in the wider class of Hausforff paratopological groups. It is proved that the closure of any subgroup of a c-compact paratopological group is again a subgroup. We present an example of a non-compact h-complete paratopological group.

Using the framework of weak selections, Nagao and Shakhmatov introduced topological king spaces, and extended the classical “King Chicken Theorem” by showing that each compact space with a continuous weak selection is a king space. They also obtained that several king spaces are compact, and raised the question whether every locally compact (or locally pseudocompact) king space must be compact. In the present paper, we settle this question in the affirmative.

A typical construction for extensions of T0 generalized topological spaces (in short, GTS) is introduced and studied. A precise description for such extensions to be μ-compact is also established. The construction of compactification of spaces in terms of grills follow as a special case of the results obtained in this paper, as μ-compactness of a GTS is a generalization of compactness of topological spaces.

We show that the component and quasi component of 0 in the space Cr(X) (C(X) with r-topology), coincide with the intersection of all principal ideals in C*(X) generated by regular elements of C*(X). Several different characterizations of the component of 0 in Cr(X) are given and using these characterizations, it turns out that Cr(X) is totally disconnected if and only if the set of non-almost P-points of βX is dense in βX. It is also shown that Cr(X) is connected or locally connected if and only if X is a pseudocompact almost P-space. We observe that most of the familiar forms of compactness and most of the countability properties of Cr(X) are equivalent to the finiteness of X. Finally, open and closed ideals in Cr(X) are investigated and it is proved that for every \({p \in X}\), clrOp = Mp if and only if X is an almost P-space.

We study embeddings in maximal pseudocompact spaces together with maximal countable compactness in the class of Tychonoff spaces. It is proved that under MA $${+\neg}$$ CH any compact space of weight $${\kappa < \mathfrak{c}}$$ is a retract of a compact maximal pseudocompact space. If κ is strictly smaller than the first weakly inaccessible cardinal, then the Tychonoff cube [0, 1]κ is maximal countably compact. However, for a measurable cardinal κ, the Tychonoff cube of weight κ is not even embeddable in a maximal countably compact space. We also show that if X is a maximal countably compact space, then the functional tightness of X is countable. It is independent of ZFC whether every compact space of countable tightness must be maximal countably compact. On the other hand, any countably compact space X with the Mazur property ( $${\equiv}$$ every real-valued sequentially continuous function on X is continuous) must be maximal countably compact. We prove that for any ω-monolithic compact space X, if C p (X) has the Mazur property, then it is a Frechet–Urysohn space.

For N-point best-packing configurations ω N on a compact metric space (A,ρ), we obtain estimates for the mesh-separation ratio γ(ω N ,A), which is the quotient of the covering radius of ω N relative to A and the minimum pairwise distance between points in ω N . For best-packing configurations ω N that arise as limits of minimal Riesz s-energy configurations as s→∞, we prove that γ(ω N ,A)≦1 and this bound can be attained even for the sphere. In the particular case when N=5 on S 2 with ρ the Euclidean metric, we prove our main result that among the infinitely many 5-point best-packing configurations there is a unique configuration, namely a square-base pyramid $\omega_{5}^{*}$ , that is the limit (as s→∞) of 5-point s-energy minimizing configurations. Moreover, $\gamma(\omega_{5}^{*},S^{2})=1$ .

We show that there exists a natural approach version of the topological Vietoris hyperspace construction [16], [17] which has several advantages over the topological version. In the first place the important structural fact that the Vietoris construction can now also be considered, not only for topological but also intrinsically for metric spaces, but equally important in the second place the fact that we can considerably strengthen fundamental classic results. In this paper we mainly pay attention to properties concerning or involving compactness. As main results, in the first place we prove that it is not merely compactness of the Vietoris hyperspace which is equivalent to compactness of the original space [3] but that actually in the approach setting the indices of compactness [7], [8], [9], [10] numerically completely coincide. In the second place the well-known result [3], [4], [15] which says that if the original space is compact metric then the Vietoris topology is metrizable by the Hausdorff metric gets strengthened in the sense that in the approach setting under the same conditions the Vietoris approach structure actually coincides with the Hausdorff metric. Classic results follow as easy corollaries. Besides these main results we also draw attention to the good functorial relationship between the Vietoris approach structures and the associated topologies.

Baireness on generalized topological spaces is introduced, and its characterizations and properties are investigated.

We deal with incompactness. Assume the existence of non-reflecting stationary set of cofinality κ. We prove that one can define a graph G whose chromatic number is >κ, while the chromatic number of every subgraph G′⫅G, |G′

We study the doubling property of binomial measures on the middle interval Cantor set. We obtain a necessary and sufficient condition that enables a binomial measure to be doubling. Then we determine those doubling binomial measures which can be extended to be doubling on [0,1]. Finally, we construct a compact set X in [0,1] and a doubling measure μ on X, such that \(\overline{F}_{X}=X\) and \({\mu|}_{E_{X}}\) is doubling on EX, where EX is the set of accumulation points of X and FX is the set of isolated points of X.

Cofinal completeness of a metric space, which is a property between completeness and compactness, can be characterized in terms of a measure of local compactness functional [7]. Using this functional, we introduce and then study the variational notion of cofinally complete subset of a metric space.

We consider sequences of functions that have in some sense a universal distribution of limit points of zeros in the complex plane. In particular, we prove that functions having universal approximation properties on compact sets with connected complement automatically have such a universal distribution of limit points. Moreover, in the case of sequences of derivatives, we show connections between this kind of universality and some rather old results of Edrei/MacLane and Polya. Finally, we show the lineability of the set what we call Jentzsch-universal power series.

The aim of this paper is to consider compactness notions by utilizing Λ-sets, V-sets, locally closed sets, locally open sets, λ-closed sets and λ-open sets. We completely characterize these variations of compactness, and also provide various interesting examples that support our results.

Notions of strongly and absolutely closed objects with respect to a closure operator X on an arbitrary category X and with respect to a subcategory Y are introduced. This yields two Galois connections between closure operators on a given category X and subclasses of X, whose fixed points are studied. A relationship with some compactness notions is shown and examples are provided.

We introduce and study a concept of neighborhoods with respect to a categorical closure operator. The concept, which is based on using pseudocomplements in subobject lattices, naturally generalizes the classical neighborhoods in topological spaces and we show that it behaves accordingly. We investigate also separation and compactness defined in a natural way by the help of the neighboorhoods introduced.

Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \colon \mathcal{M}(X) \to \R$ by $I(mu) = \int_X \int_X d(x,y) d\mu(x) d\mu(y)$, and set $M(X) = \sup I(mu)$, where $\mu$ ranges over the collection of measures in $\mathcal{M}(X)$ of total mass 1. The space $(X, d)$ is \emph{quasihypermetric} if $I(\mu) \leq 0$ for all measures $\mu$ in $\mathcal{M}(X)$ of total mass 0 and is \emph{strictly quasihypermetric} if in addition the equality $I(\mu) = 0$ holds amongst measures $\mu$ of mass 0 only for the zero measure. This paper explores the constant $M(X)$ and other geometric aspects of $X$ in the case when the space $X$ is finite, focusing first on the significance of the maximal strictly quasihypermetric subspaces of a given finite quasihypermetric space and second on the class of finite metric spaces which are $L^1$-embeddable. While most of the results are for finite spaces, several apply also in the general compact case. The analysis builds upon earlier more general work of the authors [Peter Nickolas and Reinhard Wolf, \emph{Distance geometry in quasihypermetric spaces. I}, \emph{II} and \emph{III}]., Comment: 21 pages. References [11], [12] and [13] are arXiv:0809.0740v1 [math.MG], arXiv:0809.0744v1 [math.MG] and arXiv:0809.0746v1 [math.MG], resp

We study the relationship between the product and other basic operations (namely σ, π, α and β) of generalized topologies. Also we discuss the connectedness, generalized connectedness and compactness of products of generalized topologies. It is proved that the connectedness and compactness are preserved under the product of generalized topologies, which shows that the definition of product of generalized topologies is quite reasonable.

Characterizations and properties of $$ \mathcal{I}_g $$ -closed sets and $$ \mathcal{I}_g $$ -open sets are given. A characterization of normal spaces is given in terms of $$ \mathcal{I}_g $$ -open sets. Also, it is established that an $$ \mathcal{I}_g $$ -closed subset of an $$ \mathcal{I} $$ -compact space is $$ \mathcal{I} $$ -compact.

We prove that every point-finite family of nonempty functionally open sets in a topological space X has the cardinality at most an infinite cardinal κ if and only if w(X) ≦ κ for every Valdvia compact space Y\( \subseteqq \)Cp(X). Correspondingly a Valdivia compact space Y has the weight at most an infinite cardinal κ if and only if every point-finite family of nonempty open sets in Cp(Y) has the cardinality at most κ, that is p(Cp(Y)) ≦ κ. Besides, it was proved that w(Y) = p(Cp(Y)) for every linearly ordered compact Y. In particular, a Valdivia compact space or linearly ordered compact space Y is metrizable if and only if p(Cp(Y)) = ℵ0. This gives answer to a question of O. Okunev and V. Tkachuk.

By introducing a conservative notion of functional compactness to frames and using frame-theoretic techniques, we identify two classes of topological spaces that can be densely embedded in functionally compact ones. Since θ-open elements of frames (corresponding in spaces to complements of Velicko’s [15] θ-closed subspaces) play an important role in functional compactness, we also study the concept of co-rigidity in frames and show that it is enjoyed by all θ-open elements of functionally compact frames.

Given r ∈ [0, 1] we study descriptive complexity of the set Pr (respectively Dr) of all compact sets K in the hyperspace \( \mathcal{K} \)(R) with porosity r (density r) at 0. We also show that the set NBP of all nowhere bilaterally porous compact sets in \( \mathcal{K} \)(R) is Π11-complete, and we prove a similar fact for density.

We investigate local properties of the Green function of the complement of a compact set Ein R d, d>2. We give a Wiener type characterization for the Holder continuity of the Green function, thus extending a result of L. Carleson and V. Totik. The obtained density condition is necessary, and it is sufficient as well, provided Esatisfies the cone condition. It is also shown that the Holder condition for the Green function at a boundary point can be equivalently stated in terms of the equilibrium measure and the solution to the corresponding Dirichlet problem. The results solve a long standing open problem -- raised by Maz'ja in the 1960's -- under the simple cone condition.

We generalize the classical Riesz convergence theorem to Lorentz spaces, i.e., if f,f 1 , f 2 ,.. e . L p,q such that f n → f (a.e. or in measure) and ׀׀ f n ׀׀ p,q → ׀׀ f ׀׀ p,q , then ׀׀ f n - f ׀׀ p,q → 0 as n →∞.

Three weak variants of compactness which lie strictly between compactness and quasicompactness, are introduced. Their basic properties are studied. The interplay with mapping and their direct and inverse preservation under mappings are investigated. In the process three decompositions of compactness are observed.

The following generalization of the Hahn-Mazurkiewicz theorem is proved: Let (E,e) be a locally compact locally connected metric space. Let M be a continuum in this space and let d,e∈ M. Then there is a continuous mapping f: [0,1]→E such that f(0) = d, f(1)= e and M⊂f([0,1]). Also some corollaries of this theorem are proved.

We obtain (a) necessary and sufficient conditions and (b) sufficient conditions for a compact (countably compact) set to be closed in products (sequential products) and subspaces (sequential subspaces) of normal spaces. As a consequence of these, sufficient conditions are obtained for (i) the closedness of arbitrary (countable) union of closed sets and (ii) the equality of the union of the closures and the closure of the union of arbitrary (countable) families of sets in these spaces. It is also shown that these results do not hold for quotients of even T 4,-spaces.

We prove that there are Tychonoff spaces X for which p(Cp(X)) =ϖ and Cp(X) is a Lindelof Σ-space while the network weight of X is uncountable. This answers Problem 75 from [4]. An example of a space Y is given such that p(Y)=ϖ and Cp(Y) is a Lindelof Σ-space, while the network weight of Y is uncountable. This gives a negative answer to Problem 73 from [4]. For a space X with one non-isolated point a necessary and sufficient condition in terms of the topology on X is given for Cp(X) to have countable point-finite cellularity.

Characterizations of final κ+ -compactness in countably compact spaces and in weakly κ\(\bar \theta \)‐refinable spaces are given.

We extend a theorem of Hamlett and Jankovic by proving that if a topological space (X, τ) is compact with respect to the countable extension of I, then the local function A*(I) of every subset A of X with respect to τ and I is a compact subspace with respect to the extension Ĩ in A* (I). We also give a generalized version of the Banach category theorem.

The concept of a fibrant extension of compacta can be used in the study of some properties related to different conditions of movability. For example, "empty" strong shape components of a given compact metric space X correspond to those path components of a fibrant extension X of X which do not intersect X. The fibrant extension X can be constructed as a cotelescope of an ANR-sequence associated with X. Using this representation of X, we prove the following: If a continuum is movable and virtually pointed 1-movable, then it is pointed movable. As a corollary, we get immediately that movable continua, which do not have "empty" strong shape components, are pointed movable. In particular, continua, both fibrant and movable, are of this kind. In fact, they are locally path connected approximate polyhedra.

We give a new and simple proof to show that no homeomorphism of infinite compact metric spaces is positively expansive.

We show that a bounded subset K of Lp(μ,X) is relatively norm compact if and only if K is p-uniformly integrable, scalarly relatively compact, and either tight or flatly concentrated. The scalar relative compactness can be also replaced by several oscillation criteria.

Let f be an essential mapping from a compact metric space X onto the n-dimensional disk Dn and let n ≦ 2 or dim X < 2n - 2. It is known that f is stably essential, i.e., the product mapping f × idIk of f and the identity mapping idIk on the k-dimensional cube Ik is essential for all k. In this paper it is shown that the m-fold cone mapping Cm(f) : Cm(X) → Dm+n and the m-fold suspension mapping Sm(f) : Sm(X) → Dm+n are stably essential for any m. It is also established that under the assumption mentioned above the mappings f × idIm, Cm(f) and Sm(f) are all coincidence universal for any m.

The relations between the (strong) reachable sets of the semilinear evolution equation systems x′(t) + A(t)x(t) = f(t, x(t), u(t)) + Hu(t), x′(t) + A(t)x(t) = f(t, x(t), Hu(t)) + Hu(t) on a Banach space, and their corresponding linear systems are studied. Compared with previous results, the systems considered here are more general (f is not independent of the control u), no compactness assumptions on A or f are imposed in some of our main results, and we suppose f is a set-contraction rather than Lipschitz and have less restriction on the contraction coefficient. Other kinds of conditions are involved to guarantee the approximate controllability.

We prove that every countably compact space with quasi-S1-diagonal is compact. However, it is shown that it need not be metrizable.

We call K a uniquely remotal set, if for all xCX, Q~(x) consists of exactly one element. In this case we denote by QK (x) this element of K. The following problem is called the farthest point problem. The problem as far as we know is open. Let K be a uniquely remotal set in (X, II 9 II). Is K a singleton? There are many special cases in which the problem is solved affirmatively. This is the case for finite dimensional X [I], for norm-compact K [1], for normcontinuous QK [2], in the Banach spaces co, c [7]; with a suitable renorming in all normed linear spaces [5]. In [4] we have given a positive answer in many special finite codimensional subspaces of C[0, 1], including C. In this note we give a general solution for all finite codimensional subspaces of C(12), where f2 is a compact metric space.