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Let $\kappa$ be an uncountable cardinal with $\kappa=\kappa^{{\kappa^+$'' implies the statement "every closed subset of ${}^\kappa\kappa$ is a continuous image of ${}^\kappa(\kappa^+)$'' or its negation., Comment: 30 pages

In this paper, we generalize the concept of a monotone convergence space (also called a d-space) to the setting of a Q -cotopological space, where Q is a commutative and integral quantale. We establish a D-completion for every stratified Q -cotopological space, which is a category reflection of the category S Q -CTop of stratified Q -cotopological spaces onto the full subcategory S Q -DCTop of monotone convergence Q -cotopological spaces. By introducing the notion of a tapered set, a direct characterization of the completion is obtained: the D-completion of each stratified Q -cotopological space X consists exactly of those tapered closed sets in X. We show that the D-completion can be applied to obtain a universal fuzzy directed completion of a Q -ordered set by endowing it with the Scott cotopology, taking the D-completion, and then passing to the specialization Q -order. Consequently, the category Q -DOrd of fuzzy directed complete Q -ordered sets and Scott continuous functions is reflective in the category Q -Ordσ of Q -ordered sets and Scott continuous functions.

It is known that the Hurwitz zeta-function $$\zeta(s,\alpha)$$ with transcendental or rational parameter $$\alpha$$ has a discrete universality property; i.e., the shifts $$\zeta(s+ikh,\alpha)$$ , $$k\in\mathbb N_0$$ , $$h> 0$$ , approximate a wide class of analytic functions. The case of algebraic irrational $$\alpha$$ is a complicated open problem. In the paper, some progress in this problem is achieved. It is proved that there exists a nonempty closed set $$F_{\alpha,h}$$ of analytic functions such that the functions in $$F_{\alpha,h}$$ are approximated by the above shifts. Also, the case of certain compositions $$\Phi(\zeta(s,\alpha))$$ is discussed.

الفكرة الرئيسية لهذا العمل الحالي هي تقديم أنواع جديدة من المجموعات فوق المغلقة في فضاءات فوق طوبولوجية تسمى supra -closed (باختصار ، مغلقة). علاوة على ذلك ، قارن هذه الفئة من المجموعات بأنواع أخرى من المجموعات أعلاه في الفضاءات فوق الطوبولوجية. دراسة وإثبات بعض صفاتهم.

In this paper pgrw-locally closed set, pgrw-locally closed*-set and pgrw-locally closed**-set are introduced. A subset A of a topological space (X,t) is called pgrw-locally closed (pgrw-lc) if A=GÇF where G is a pgrw-open set and F is a pgrw-closed set in (X,t). A subset A of a topological space (X,t) is a pgrw-lc* set if there exist a pgrw-open set G and a closed set F in X such that A= GÇF. A subset A of a topological space (X,t) is a pgrw-lc**-set if there exists an open set G and a pgrw-closed set F such that A=GÇF. The results regarding pgrw-locally closed sets, pgrw-locally closed* sets, pgrw-locally closed** sets, pgrw-lc-continuous maps and pgrw-lc-irresolute maps and some of the properties of these sets and their relation with other lc-sets are established.

In this paper, it is shown that for a minimal system (X, G), if H is a normal subgroup of G with finite index n, then X can be decomposed into n components of closed sets such that each component is minimal under H-action. Meanwhile, we prove that for a residual set of points in a minimal system with finitely many commuting homeomorphisms, the set of return times to any non-empty open set contains arbitrarily long geometric progressions in multidimension, extending a previous result by Glasscock, Koutsogiannis and Richter.

A set of non-negative integral vectors is said to be right-closed if the presence of a vector in the set implies all term-wise larger vectors also belong to the set. A set of markings is control invariant with respect to a Petri Net (PN) structure if the firing of any uncontrollable transition at any marking in this set results in a new marking that is also in the set. Every right-closed set of markings has a unique supremal control invariant subset, which is the largest subset that is control invariant with respect to the PN structure. This subset is not necessarily right-closed. In this paper, we present an algorithm that computes the supremal right-closed control invariant subset of a right-closed of markings with respect to an arbitrary PN structure. This set plays a critical role in the synthesis of Liveness Enforcing Supervisory Policies (LESPs) for a class of PN structures, and consequently, the proposed algorithm plays a key role in the synthesis of LESPs for this class of PN structures.

In this paper, we introduce the notions of Belluce lattice associated with a bounded $BCK$-algebra and reticulation of a bounded $BCK$-algebra. To do this, first, we define the operations $curlywedge ,$ $curlyvee $ and $sqcup $ on $BCK$-algebras and we study some algebraic properties of them. Also, for a bounded $BCK$-algebra $A$ we define the Zariski topology on $ Spec(A)$ and the induced topology $tau _{A,Max(A)}$ on $Max(A)$. We prove $(Max(A),tau_{A,Max(A)})$ is a compact topological space if $A$ has Glivenko property. Using the open and the closed sets of $Max(A)$, we define a congruence relation on a bounded $BCK$-algebra $A$ and we show $L_{A}$, the quotient set, is a bounded distributive lattice. We call this lattice the Belluce lattice associated with $A.$ Finally, we show $(L_{A},p_{A})$ is a reticulation of $A$ (in the sense of Definition ref{d7}) and the lattices $L_{A}$ and $S_{A}$ are isomorphic.

In this paper multioperations defined on a two-element set and their closure operator based on composition operator and the equality predicate branching operator is considered. The composition operator is based on union of sets. The classification of multioperations based on their membership in precomplete sets has been obtained. It is shown that the number of equivalence classes is 129. All types of bases are described and it is proved that the maximum cardinality of a basis is 4.

We give a classification of irreducible representations of generalized Heisenberg groups K n ( F q ) , n ≥ 5 , which is the pattern group associated to the closed set { ( 1 , i ) , ( 2 , j ) , ...

Let X be a set and f : X →X be a map. We denote by P(f) the topology dened on X whose closed sets are the subsets A of X with f(A) A⊆A topology on X is said to be a primal topology, if it is a P(f) for some map f. Our aim here is to characterize when the product of an arbitrary family of topological spaces is a primal space.

An unconstrained nonlinear binary optimization problem of selecting a maximum expected value subset of items is considered. Each item is associated with a profit and probability. Each of the items succeeds or fails independently with the given probabilities, and the profit is obtained in the event that all selected items succeed. The objective is to select a subset that maximizes the total value times the product of probabilities of the chosen items. The problem is proven NP-hard by a nontrivial reduction from subset sum. Then, after proving bounds on the probabilities of items selected in an optimal solution, we develop a fully polynomial time approximation scheme (FPTAS) for this problem. Next we consider an extension to maximizing the same objective over a more general downward closed set system. Here the problem is reduced to the solution of a sequence of constrained linear problems. This method is then used to propose a polynomial time approximation scheme (PTAS) for maximum expected value all-or-nothing matching in a graph and all-or-nothing matroid intersection using results for the corresponding constrained linear problem variants.

Let $X\in\text{Alex}\,^n(-1)$ be an $n$-dimensional Alexandrov space with curvature $\ge -1$. Let the $r$-scale $(k,\epsilon)$-singular set $\mathcal S^k_{\epsilon,\,r}(X)$ be the collection of $x\in X$ so that $B_r(x)$ is not $\epsilon r$-close to a ball in any splitting space $\mathbb R^{k+1}\times Z$. We show that there exists $C(n,\epsilon)>0$ and $\beta(n,\epsilon)>0$, independent of the volume, so that for any disjoint collection $\big\{B_{r_i}(x_i):x_i\in \mathcal S_{\epsilon,\,\beta r_i}^k(X)\cap B_1, \,r_i\le 1\big\}$, the packing estimate $\sum r_i^k\le C$ holds. Consequently, we obtain the Hausdorff measure estimates $\mathcal H^k(\mathcal S^k_\epsilon(X)\cap B_1)\le C$ and $\mathcal H^n\big(B_r (\mathcal S^k_{\epsilon,\,r}(X))\cap B_1(p)\big)\leq C\,r^{n-k}$. This answers an open question asked by Kapovitch and Lytchak. We also show that the $k$-singular set $\mathcal S^k(X)=\underset{\epsilon>0}\cup\left(\underset{r>0}\cap\mathcal S^k_{\epsilon,\,r}\right)$ is $k$-rectifiable and construct examples to show that such a structure is sharp. For instance, in the $k=1$ case we can build for any closed set $T\subseteq \mathbb S^1$ and $\epsilon>0$ a space $Y\in\text{Alex}^3(0)$ with $\mathcal S^{1}_\epsilon(Y)=\phi(T)$, where $\phi\colon\mathbb S^1\to Y$ is a bi-Lipschitz embedding. Taking $T$ to be a Cantor set it gives rise to an example where the singular set is a $1$-rectifiable, $1$-Cantor set with positive $1$-Hausdorff measure., Comment: 28 pages

A Semmes surface in the Heisenberg group is a closed set $S$ that is upper Ahlfors-regular with codimension one and satisfies the following condition, referred to as Condition B. Every ball $B(x,r)$ with $x \in S$ and $0 < r < \operatorname{diam} S$ contains two balls with radii comparable to $r$ which are contained in different connected components of the complement of $S$. Analogous sets in Euclidean spaces were introduced by Semmes in the late $80$'s. We prove that Semmes surfaces in the Heisenberg group are lower Ahlfors-regular with codimension one and have big pieces of intrinsic Lipschitz graphs. In particular, our result applies to the boundary of chord-arc domains and of reduced isoperimetric sets., 39 pages, 4 figures

If $f:X \rightarrow X$ is a function, the associated functional Alexandroff topology on X is the topology Pf whose closed sets are $\{A \subseteq X : f(A) \subseteq A\}$ . We present a characterization of functional Alexandroff topologies on a finite set X and show that the collection FA(X) of all functional Alexandroff topologies on a finite set X, ordered by inclusion, is a complemented lattice.

International audience; Given an implicational base, a well-known representation for a closure system, an inconsistency binary relation over a finite set, we are interested in the problem of enumerating all maximal consistent closed sets (denoted by MCCEnum for short). We show that MCCEnum cannot be solved in output-polynomial time unless P = NP, even for lower bounded lattices. We give an incremental-polynomial time algorithm to solve MCCEnum for closure systems with constant Carathéodory number. Finally we prove that in biatomic atomistic closure systems MCCEnum can be solved in output-quasipolynomial time if minimal generators obey an independence condition, which holds in atomistic modular lattices. For closure systems closed under union (i.e., distributive), MCCEnum is solved by a polynomial delay algorithm [22, 25].

The aim of this paper is to present separation theorems for two disjoint closed sets, without convexity condition. First, a separation theorem for a given closed cone and a point outside from this cone, is proved and then it is used to prove a separation theorem for two disjoint sets. Illustrative examples are provided to highlight the important aspects of these theorems. An application to optimization is also presented to prove optimality condition for a nonconvex optimization problem.

A family of finite sets is called union-closed if it contains the union of any two sets in it. The Union-Closed Sets Conjecture of Frankl from 1979 states that each union-closed family contains an element that belongs to at least half of the members of the family. In this paper, we study structural properties of union-closed families. It is known that under the inclusion relation, every union-closed family forms a lattice. We call two union-closed families isomorphic if their corresponding lattices are isomorphic. Let $${{\mathcal {F}}}$$ be a union-closed family and $$\bigcup _{F\in {\mathcal {F}}} F$$ be the universe of $${\mathcal {F}}$$. Among all union-closed families isomorphic to $${{\mathcal {F}}}$$, we develop an algorithm to find one with a maximum universe, and an algorithm to find one with a minimum universe. We also study properties of these two extremal union-closed families in connection with the Union-Closed Set Conjecture. More specifically, a lower bound of sizes of sets of a minimum counterexample to the dual version of the Union-Closed Sets Conjecture is obtained.

This paper pursues an investigation on the lattices of irreducibly-derived closed sets initiated by Zhao and Ho (2015). This time we focus the closed set lattice arising from the irreducibly-derived topology of Scott topology. For a poset X, the set Γ S I ( X ) of all irreducibly-derived Scott-closed sets (for short, SI-closed sets) ordered by inclusion forms a complete lattice. We introduce the notions of C S I -continuous posets and C S I -prealgebraic posets and study their properties. We also introduce the SI-dominated posets and show that for any two SI-dominated posets X and Y, X ≅ Y if and only if the SI-closed set lattices above them are isomorphic. At last, we show that the category of strong complete posets with SI-continuous maps is Cartesian-closed.

We consider a class of compact positive operators $$L:X\rightarrow X$$ given by $$(Lx)(t)=\int ^t_{\eta (t)}x(s)\,ds$$ , acting on the space X of continuous $$2\pi $$ -periodic functions x. Here $$\eta $$ is continuous with $$\eta (t)\le t$$ and $$\eta (t+2\pi )=\eta (t)+2\pi $$ for all $$t\in \mathbf{R}$$ . We obtain necessary and sufficient conditions for the spectral radius of L to be positive, in which case a nonnegative eigensolution to the problem $$\kappa x=Lx$$ exists for some $$\kappa >0$$ (equal to the spectral radius of L) by the Krein–Rutman theorem. If additionally $$\eta $$ is analytic, we study the set $${\mathcal {A}}\subseteq \mathbf{R}$$ of points t at which x is analytic; in general $${\mathcal {A}}$$ is a proper subset of $$\mathbf{R}$$ , although x is $$C^\infty $$ everywhere. Among other results, we obtain conditions under which the complement $${\mathcal {N}}=\mathbf{R}{\setminus }{\mathcal {A}}$$ of $${\mathcal {A}}$$ is a generalized Cantor set, namely, a nonempty closed set with empty interior and no isolated points. The proofs of this and of other such results depend strongly on the dynamical properties of the map $$t\rightarrow \eta (t)$$ .

In this paper, we defined -open set by using s-operation and -closed set, then by using -open set, we define -closed set. In addition we define -closure of subset of ( ) and -interior of subset of by using -closed set and -open set respectively. Furthermore we introduce and discuss minimal -open sets in topological spaces. We establish some basic properties of minimal -open. We obtain an application of a theory of minimal -open sets and define a -locally finite space then we prove, Let be a -locally finite space and a nonempty -open set. Then there exists at least one (finite) minimal -open set such that where is semi-regular.

If f:X→X is a map, then the family {A⊆X:f(A)⊆A} gives the closed sets of a topology P(f) on X called a functionally Alexandroff topology. If (X,P(f)) is a connected homogeneous functionally...

Let $X$ be a first countable space which admits a non-trivial convergent sequence and let $\mathcal{I}$ be an analytic P-ideal. First, it is shown that the sets of $\mathcal{I}$-limit points of all sequences in $X$ are closed if and only if $\mathcal{I}$ is also an $F_\sigma$-ideal. Moreover, let $(x_n)$ be a sequence taking values in a Polish space without isolated points. It is known that the set $A$ of its statistical limit points is an $F_\sigma$-set, the set $B$ of its statistical cluster points is closed, and that the set $C$ of its ordinary limit points is closed, with $A\subseteq B\subseteq C$. It is proved the sets $A$ and $B$ own some additional relationship: indeed, the set $S$ of isolated points of $B$ is contained also in $A$. Conversely, if $A$ is an $F_\sigma$-set, $B$ is a closed set with a subset $S$ of isolated points such that $B\setminus S\neq \emptyset$ is regular closed, and $C$ is a closed set with $S\subseteq A\subseteq B\subseteq C$, then there exists a sequence $(x_n)$ for which: $A$ is the set of its statistical limit points, $B$ is the set of its statistical cluster points, and $C$ is the set of its ordinary limit points. Lastly, we discuss topological nature of the set of $\mathcal{I}$-limit points when $\mathcal{I}$ is neither $F_\sigma$- nor analytic P-ideal., Comment: 15 pages, comments are welcome

Given a function $f\colon \mathbb{R}\to \mathbb{R}$, the so-called "little lip" function $\operatorname{lip} f$ is defined as follows: \begin{equation*} \operatorname{lip} f(x)=\liminf_{r{\scriptscriptstyle \searrow} 0}\sup_{|x-y|\le r} \frac{|f(y)-f(x)|}{r}. \end{equation*} We show that if $f$ is continuous on $\mathbb{R}$, then the set where $\operatorname{lip} f$ is infinite is a countable union of a countable intersection of closed sets (that is an $F_{\sigma \delta}$ set). On the other hand, given a countable union of closed sets $E$, we construct a continuous function $f$ such that $\operatorname{lip} f$ is infinite exactly on $E$. A further result is that for the typical continuous function $f$ on the real line $\operatorname{lip} f$ vanishes almost everywhere., Comment: 27 pages, 3 figures. We updated affiliations and the acknowledgements section and reformatted the paper. It is accepted for publication in Studia Mathematica

We make use of the fact that a two-sided whole-plane Schramm–Loewner evolution (SLE $$_\kappa $$ ) curve $$\gamma $$ for $$\kappa \in (0,8)$$ from $$\infty $$ to $$\infty $$ through 0 may be parametrized by its d-dimensional Minkowski content, where $$d=1+\frac{\kappa }{8}$$ , and become a self-similar process of index $$\frac{1}{d}$$ with stationary increments. We prove that such $$\gamma $$ is locally $$\alpha $$ -Holder continuous for any $$\alpha

In this chapter, we discuss some generalizations of total graphs of commutative rings. Actually, total graphs of commutative rings are generalized through ideals, multiplicatively closed sets, and multiplicative-prime sets. Further, we discuss the concept of total graphs corresponding to modules and commutative semirings.

In this article we treat two closely related problems: 1) the upper semi-continuity property for Almgren minimal sets in regions with regular boundary; and 2) the uniqueness property for all the $2$-dimensional minimal cones in $\mathbb R^3$. ¶ Given an open set $\Omega\subset\mathbb R^n$, a closed set $E\subset \Omega$ is said to be Almgren minimal of dimension $d$ in $\Omega$ if it minimizes the $d$-Hausdorff measure among all its Lipschitz deformations in $\Omega$. We say that a $d$-dimensional minimal set $E$ in an open set $\Omega$ admits upper semi-continuity if, whenever $\{f_n(E)\}_n$ is a sequence of deformations of $E$ in $\Omega$ that converges to a set $F$, then we have ${\mathcal H}^d(F)\ge \limsup_n {\mathcal H}^d(f_n(E))$. This guarantees in particular that $E$ minimizes the $d$-Hausdorff measure, not only among all its deformations, but also among limits of its deformations. ¶ As proved in [19], when several $2$-dimensional minimal cones are all translational and sliding stable, and admit the uniqueness property, then their almost orthogonal union stays minimal. As a consequence, the uniqueness property obtained in the present paper, together with the translational and sliding stability properties proved in [18] and [20] permit us to use all known $2$-dimensional minimal cones in $\mathbb R^n$ to generate new families of minimal cones by taking their almost orthogonal unions. ¶ The upper semi-continuity property is also helpful in various circumstances: when we have to carry on arguments using Hausdorff limits and some properties do not pass to the limit, the upper semi-continuity can serve as a link. As an example, it plays a very important role throughout [19].

A Hausdorff topological space X is called superconnected (resp. coregular) if for any nonempty open sets U 1 , … U n ⊆ X , the intersection of their closures U ‾ 1 ∩ … ∩ U ‾ n is not empty (resp. the complement X ∖ ( U ‾ 1 ∩ … ∩ U ‾ n ) is a regular topological space). A canonical example of a coregular superconnected space is the projective space Q P ∞ of the topological vector space Q ω = { ( x n ) n ∈ ω ∈ Q ω : | { n ∈ ω : x n ≠ 0 } | ω } over the field of rationals Q . The space Q P ∞ is the quotient space of Q ω ∖ { 0 } ω by the equivalence relation x ∼ y iff Q ⋅ x = Q ⋅ y . We prove that every countable second-countable coregular space is homeomorphic to a subspace of Q P ∞ , and a topological space X is homeomorphic to Q P ∞ if and only if X is countable, second-countable, and admits a decreasing sequence of closed sets ( X n ) n ∈ ω such that (i) X 0 = X , ⋂ n ∈ ω X n = ∅ , (ii) for every n ∈ ω and a nonempty relatively open set U ⊆ X n the closure U ‾ contains some set X m , and (iii) for every n ∈ ω the complement X ∖ X n is a regular topological space. Using this topological characterization of Q P ∞ we find topological copies of the space Q P ∞ among quotient spaces, orbit spaces of group actions, and projective spaces of topological vector spaces over countable topological fields.

The family of \(\mathcal {T}\)-closed sets have been considered by several authors, for example (FitzGerald and Swingle, Fund. Math., 61:33–50, 1967) and (E. J. Vought, Pacific J. Math., 54: 253–261, 1974). We introduce the family of \(\mathcal {T}\)-closed sets of a continuum X and present its main properties. We also give a characterization of \(\mathcal {T}\)-closed sets. We consider minimal \(\mathcal {T}\)-closed sets and the set function \(\mathcal {T}^\infty \). We introduce the \(\mathcal {T}\)-growth bound of a continuum.

We prove basic results about the set function \({\mathcal {T}}\) defined by F. Burton Jones (Amer J Math 70:403–413, 1948) to study the properties of metric continua. We define this function on compacta, and then we concentrate on continua. In particular, we present some of the well known properties (such as connectedness im kleinen, local connectedness, semi-local connectedness, etc.) using the set function \({\mathcal {T}}\). The notion of aposyndesis was the main motivation of Jones to define this function. We present some properties of a continuum when it is \({\mathcal {T}}\)-symmetric and \({\mathcal {T}}\)-additive. We give properties of continuum on which \({\mathcal {T}}\) is idempotent, idempotent on closed sets and idempotent on contina. We also present results about the set functions \({\mathcal {T}}^n\), when \(n\in {\mathbb {N}}\).

A poset P forms a locally compact \(T_0\)-space in its Alexandroff topology. We consider the hit-or-miss topology on the closed sets of P and the associated Fell compactification of P. We show that the closed sets of P with the hit-or-miss topology is the Priestley space of the bounded distributive lattice freely generated by the order dual of P. The Fell compactification of H(P) is shown to be the Priestley space of a sublattice of the upsets of P consisting of what we call Fell upsets. These are upsets that are finite unions of those obtained as upper bounds of finite subsets of P. The restriction of the hit topology to H(P) is a stable compactification of P. When P is a chain, we show that this is the least stable compactification of P.

Altres ajuts: SEV-2015-0563 We present necessary and sufficient optimality conditions for the minimization of pseudoconvex functions over convex intersections of non necessarily convex sets. To this aim, we use the notion of local normal cone to a closed set at a point, due to Linh and Penot (SIAM J Optim 17:500-510, 2006). The technique we use to obtain the optimality conditions is based on the so called canonical representation of a closed set by means of its associated oriented distance function.

Special α-limit sets (sα-limit sets) combine together all accumulation points of all backward orbit branches of a point x under a noninvertible map. The most important question about them is whether or not they are closed. We challenge the notion of sα-limit sets as backward attractors for interval maps by showing that they need not be closed. This disproves a conjecture by Kolyada, Misiurewicz, and Snoha. We give a criterion in terms of Xiong’s attracting centre that completely characterizes which interval maps have all sα-limit sets closed, and we show that our criterion is satisfied in the piecewise monotone case. We apply Blokh’s models of solenoidal and basic ω-limit sets to solve four additional conjectures by Kolyada, Misiurewicz, and Snoha relating topological properties of sα-limit sets to the dynamics within them. For example, we show that the isolated points in a sα-limit set of an interval map are always periodic, the non-degenerate components are the union of one or two transitive cycles of intervals, and the rest of the sα-limit set is nowhere dense. Moreover, we show that sα-limit sets in the interval are always both F σ and G δ . Finally, since sα-limit sets need not be closed, we propose a new notion of β-limit sets to serve as backward attractors. The β-limit set of x is the smallest closed set to which all backward orbit branches of x converge, and it coincides with the closure of the sα-limit set. At the end of the paper we suggest several new problems about backward attractors.

The generalized closed set plays an important role in the study of fuzzy topological spaces, and this paper is a prolongation of the idea of \(g^*\)-fuzzy closed set in the light of fuzzy \(\gamma ^*\)-open set in the same environment. In particular, this paper presents three different types of \(g^*\)-fuzzy closed sets, namely \(\gamma ^*\)-\(g^*\)-fuzzy closed set, \(g^*\)-\(\gamma ^*\)-fuzzy closed set, and \(\gamma ^*\)-\(g^*\)-\(\gamma ^*\) fuzzy closed set via fuzzy \(\gamma ^*\)-open set in fuzzy topological spaces. Also, we establish the interrelationships among these newly defined fuzzy sets with the existing ones. As an application, we introduce and study some new classes of spaces called fuzzy \(\gamma ^*\)-\({T_{1/2}}^*\) spaces, fuzzy \({T_{1/2}}^*\)-\(\gamma ^*\) space and fuzzy \(\gamma ^*\)-\({T_{1/2}}^*\)-\(\gamma ^*\) space.

We show some new results about tilings in Banach spaces. A tiling of a Banach space $X$ is a covering by closed sets with non-empty interior so that they have pairwise disjoint interiors. If moreover the tiles have inner radii uniformly bounded from below, and outer radii uniformly bounded from above, we say that the tiling is normal. In 2010 Preiss constructed a convex normal tiling of the separable Hilbert space. For Banach spaces with Schauder basis we will show that Preiss' result is still true with starshaped tiles instead of convex ones. Also, whenever $X$ is uniformly convex we give precise constructions of convex normal tilings of the unit sphere, the unit ball or in general of any convex body., Comment: 12 pages, 2 figures

In this treatise, we propose a new type of closed set in fuzzy topological spaces called \(g^*\) fuzzy closed set, which is lying in between the fuzzy closed set and the generalized fuzzy closed set. Also, we study another class of fuzzy sets called \(\theta \)-\(g^*\) fuzzy closed sets which is weaker than \(\theta \)- fuzzy closed sets but stronger than \(\theta \)-g fuzzy closed sets and an interrelationships among these newly defined fuzzy closed sets along with the existing generalized fuzzy closed sets are established. Furthermore, the idea of fuzzy \(g^*\)-connectedness is introduced in the light of \(g^*\)-fuzzy closed sets. Finally, we define \({T_{1/2}}^*\)-space, \(^*T_{1/2}\)-space, \({_\theta T_{1/2}}^*\)-space and \({^*}_\theta T_{1/2}\)-space and some applications of these newly defined spaces are discussed.

A compact topological space X is spectral if it is sober (i.e., every irreducible closed set is the closure of a unique singleton) and the compact open subsets of X form a basis of the topology of X, closed under finite intersections. Theorem. A topological space X is homeomorphic to the spectrum of some countable Abelian {\ell}-group with unit (resp., MV-algebra) iff X is spectral, has a countable basis of open sets, and for any points x and y in the closure of a singleton {z}, either x is in the closure of {y} or y is in the closure of {x}. We establish this result by proving that a countable distributive lattice D with zero is isomorphic to the lattice of all principal ideals of an Abelian {\ell}-group (we say that D is {\ell}-representable) iff for all a, b $\in$ D there are x, y $\in$ D such that a $\lor$ b = a $\lor$ y = b $\lor$ x and x $\land$ y = 0. On the other hand, we construct a non-{\ell}-representable bounded distributive lattice, of cardinality $\aleph$ 1 , with an {\ell}-representable countable L$\infty, \omega$-elementary sublattice. In particular, there is no characterization, of the class of all {\ell}-representable distributive lattices, in arbitrary cardinality, by any class of L$\infty, \omega$ sentences., Comment: Misprints v2: In Example 7.1, (a-mb)\wedge(b-mc) \leq 0 (i.e., \wedge instead of \vee).In Corollary 8.6, X, Y^-, and Y^+ are just elements of \Op(\mathcal{H}) (not necessarily basic open)

A (convex) polytope $P$ is said to be $2$-level if for every direction of hyperplanes which is facet-defining for $P$, the vertices of $P$ can be covered with two hyperplanes of that direction. The study of these polytopes is motivated by questions in combinatorial optimization and communication complexity, among others. In this paper, we present the first algorithm for enumerating all combinatorial types of $2$-level polytopes of a given dimension $d$, and provide complete experimental results for $d \leqslant 7$. Our approach is inductive: for each fixed $(d-1)$-dimensional $2$-level polytope $P_0$, we enumerate all $d$-dimensional $2$-level polytopes $P$ that have $P_0$ as a facet. This relies on the enumeration of the closed sets of a closure operator over a finite ground set. By varying the prescribed facet $P_0$, we obtain all $2$-level polytopes in dimension $d$., Comment: 25 pages, 10 figures, 3 tables

For any ideal of closed sets in X, let be the family of those functions in C(X) whose support lie on . Further let contain precisely those functions f in C(X) for which for each ϵ > 0, {x ∈ X: |f (x)| ≥ ϵ} is a member of . Let stand for the set of all those points p in βX at which the stone extension f∗ for each f in is real valued. We show that each realcompact space lying between X and βX is of the form if and only if X is pseudocompact. We find out conditions under which an arbitrary product of spaces of the form locally- or almost locally-, becomes a space of the same form. We further show that is a free ideal (essential ideal) of C(X) if and only if is a free ideal (essential ideal) of when and only when X is locally- (almost locally-). We address the problem, when does or become identical to the socle of the ring C(X). The results obtained turn out to imply a special version of the fact obtained by Azarpanah corresponding to the choice ≡ the ideal of compact sets in X. Finally we observe tha...

A space X is said to be quasi-Rothberger if for each closed set F ⊂ X and each sequence { U n : n ∈ N } of covers of F by sets open in X, there is a U n ∈ U n for each n ∈ N such that F ⊂ ⋃ n ∈ N U n ‾ . In this article, we give necessary and sufficient conditions of the quasi-Rothberger property of linearly ordered spaces.

The space clos(X) of all nonempty closed subsets of an unbounded metric space X is considered. The space clos(X) is endowed with a metric in which a sequence of closed sets converges if and only if the distances from these sets to a fixed point θ are bounded and, for any r, the sequence of the unions of the given sets with the exterior balls of radius r centered at θ converges in the Hausdorff metric. The metric on clos(X) thus defined is not equivalent to the Hausdorff metric, whatever the initial metric space X. Conditions for a set to be closed, totally bounded, or compact in clos(X) are obtained; criteria for the bounded compactness and separability of clos(X) are given. The space of continuous maps from a compact space to clos(X) is considered; conditions for a set to be totally bounded in this space are found.

In this paper we have introduced the notion of $$\gamma _{_\mu }$$ -open sets and $$\gamma _{_\mu }$$ -closed sets in a GTS $$(X,\mu )$$ , where $$\gamma _{_\mu }$$ is a mapping on $$\mu $$ to $$\mathcal P(X)$$ to introduce the notion of minimal $$\gamma _{_\mu }$$ -open sets and studied some of its properties. As an application, we have obtained a sufficient condition for a $$\gamma _{_\mu }$$ -locally finite GTS satisfying certain conditions to be a $$\gamma _{_\mu }$$ pre- $$T_{_2}$$ space.

Given a topological space X = (X, T), we show in the Zermelo-Fraenkel set theory ZF that:(i) Every locally finite family of open sets of X is finite iff every pairwise disjoint, locally finite family of open sets is finite.(ii) Every locally finite family of subsets of X is finite iff every pairwise disjoint,locally finite family of subsets of X is finite iff every locally finite family of closed subsets of X is finite.(iii) The statement \every locally finite family of closed sets of X is finite" implies the proposition \every locally finite family of open sets of X is finite". The converse holds true in case X is T4 and the countable axiom of choice holds true.We also show:(iv) It is relatively consistent with ZF the existence of a non countably compact T1 space such that every pairwise disjoint locally finite family of closed subsets is finite but some locally finite family of subsets is infinite.(v) It is relatively consistent with ZF the existence of a countably compact T4 space including an infinite pairwise disjoint locally finite family of open (resp. closed) sets.Mathematics Subject Classification (2010): E325, 54D30.Keywords: Axiom of choice, countably compact, pseudocompact and lightly compact topological spaces

Let F be a closed subset of ℝn and let P(x) denote the metric projection (closest point mapping) of x ∈ ℝn onto F in lp-norm. A classical result of Asplund states that P is (Frechet) differentiable almost everywhere (a.e.) in ℝn in the Euclidean case p = 2. We consider the case 2 < p < ∞ and prove that the ith component Pi(x) of P(x) is differentiable a.e. if Pi(x) 6= xi and satisfies Holder condition of order 1/(p−1) if Pi(x) = xi.

We study the number of solutions N(B,F) of the diophantine equation n1n2 = n3n4, where 1 ≤ n1 ≤ B, 1 ≤ n3 ≤ B, n2, n4 ∈ F and F ⊂ [1,B] is a factor closed set. We study more particularly the case when F = {m = $$p_1^{{\varepsilon _1}}$$ · · · $$p_k^{{\varepsilon _k}}$$ , ej∈ {0, 1}, 1 ≤ j ≤ k}, p1,..., pk being distinct prime numbers.

The catenary data of a matroid M of rank r on n elements is the vector ( ν ( M ; a 0 , a 1 , … , a r ) ) , indexed by compositions ( a 0 , a 1 , … , a r ) , where a 0 ≥ 0 , a i > 0 for i ≥ 1 , and a 0 + a 1 + ⋯ + a r = n , with the coordinate ν ( M ; a 0 , a 1 , … , a r ) equal to the number of maximal chains or flags ( X 0 , X 1 , … , X r ) of flats or closed sets such that X i has rank i, | X 0 | = a 0 , and | X i − X i − 1 | = a i . We show that the catenary data of M contains the same information about M as its G -invariant, which was defined by H. Derksen (2009) [9] . The Tutte polynomial is a specialization of the G -invariant. We show that many known results for the Tutte polynomial have analogs for the G -invariant. In particular, we show that for many matroid constructions, the G -invariant of the construction can be calculated from the G -invariants of the constituents and that the G -invariant of a matroid can be calculated from its size, the isomorphism class of the lattice of cyclic flats with lattice elements labeled by the rank and size of the underlying set. We also show that the number of flats and cyclic flats of a given rank and size can be derived from the G -invariant, that the G -invariant of M is reconstructible from the deck of G -invariants of restrictions of M to its copoints, and that, apart from free extensions and coextensions, one can detect whether a matroid is a free product from its G -invariant.