It is known that if a topological space X is locally compact then the Kuratowski convergence on t... more It is known that if a topological space X is locally compact then the Kuratowski convergence on the closed subsets of X is topological. We show that the converse is true, provided that X is quasi-sober.
In this paper we will prove that simply disconnected tournaments of order at least 7 are reconstr... more In this paper we will prove that simply disconnected tournaments of order at least 7 are reconstructible, i.e. they are determined (up to isomorphisms) by their subtournaments.
Rendiconti dell'Istituto di Matematica dell'Universita di Trieste
In this note we continue the investigation carried out in [Topology Appl. 65, No. 1, 101-104 (199... more In this note we continue the investigation carried out in [Topology Appl. 65, No. 1, 101-104 (1995; Zbl 0828.54024)]. We show that every weighted quasi-metric space can be identified with a subspace of a space of some canonical type, which is constructed from a metric space. We also present a very simple method to construct a weighted quasi-metric space, as the graph of a function defined on a metric space, and show that every weighted quasi-metric space arises in this way. Similar results may be obtained if we drop the requirement that the weight function have nonnegative values.
We show that every quasi-metric space is isomorphic to a subspace of the hyperspace of a suitable... more We show that every quasi-metric space is isomorphic to a subspace of the hyperspace of a suitable metric space, endowed with the Hausdorff quasi-metric. Therefore a topological space is quasi-metrizable if and only if it can be embedded in a Hausdorff quasi-metric hyperspace.
A structure called d 0 -algebra is defined and studied, which leads to a new description of D-lat... more A structure called d 0 -algebra is defined and studied, which leads to a new description of D-lattices and generalized D-lattices by means of only one totally defined operation. c 2011 Mathematical Institute Slovak Academy of Sciences 2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 03G05; Secondary 03G25, 81P10. K e y w o r d s: set-difference, Boolean algebra, D-lattice, D-poset, generalized D-lattice, congruence, ideal.
We prove an algebraic and a topological decomposition theorem for complete D-lattices (i.e., latt... more We prove an algebraic and a topological decomposition theorem for complete D-lattices (i.e., lattice-ordered effect algebras). As a consequence, we obtain a Hammer-Sobczyk type decomposition theorem for modular measures on D-lattices.
ABSTRACT Bornological convergence is a generalization of the well known Attouch--Wets convergence... more ABSTRACT Bornological convergence is a generalization of the well known Attouch--Wets convergence. Our aim is to compare lower, upper and ``two-sided'' convergences generated by two compatible uniformities and two arbitrary bornologies. Moreover, the comparison of convergences induced by bounded-proximal topologies is characterized.
Given a Hausdorff space X, we calculate the tightness and the character of the hyperspace CL ∅ (X... more Given a Hausdorff space X, we calculate the tightness and the character of the hyperspace CL ∅ (X) of X, endowed with either the co-compact or the lower Vietoris topology, and give some estimates for the tightness of CL ∅ (X), endowed with the Fell topology.
Let L be a D-lattice, i.e. a lattice ordered effect algebra, and let BV be the Banach space of al... more Let L be a D-lattice, i.e. a lattice ordered effect algebra, and let BV be the Banach space of all real-valued functions of bounded variation on L (vanishing at 0) endowed with the variation norm. We prove the existence of a continuous Aumann-Shapley value ϕ on bv NA, the subspace of BV spanned by all functions of the form f • µ, where µ : L → [0, 1] is a nonatomic σ-additive modular measure and f : [0, 1] → R is of bounded variation and continuous at 0 and at 1. Mathematics Subject Classification (2000). 28A12, 06C15, 91A12. A. Basile et al. Positivity (D1) a b is defined if and only if b ≤ a; (D2) If b = b c. Vol. 12 (2008) On the Aumann-Shapley value 615
In 1986, A. Basile and H. Weber proved that every countable family M of σ-additive measures on a ... more In 1986, A. Basile and H. Weber proved that every countable family M of σ-additive measures on a σcomplete Boolean ring R admits a dense G δ -set of separating points, with respect to a suitable topology, in the following two cases: either (i) each element of M is s-bounded and, whenever λ, ν ∈ M are distinct, the quotient of R modulo N (λ − ν) is infinite, or (ii) each element of M is continuous.
Beer and Di Concilio have given necessary and sufficient conditions for a two-sided Attouch--Wets... more Beer and Di Concilio have given necessary and sufficient conditions for a two-sided Attouch--Wets topology to contain another on the hyperspace of non-empty closed subsets of a metrizable space as determined by metrics compatible with the topology. In the present paper, we characterize comparability of lower Attouch--Wets topologies as determined by compatible metrics.
It is known that if a topological space X is locally compact then the Kuratowski convergence on t... more It is known that if a topological space X is locally compact then the Kuratowski convergence on the closed subsets of X is topological. We show that the converse is true, provided that X is quasi-sober.
In this paper we will prove that simply disconnected tournaments of order at least 7 are reconstr... more In this paper we will prove that simply disconnected tournaments of order at least 7 are reconstructible, i.e. they are determined (up to isomorphisms) by their subtournaments.
Rendiconti dell'Istituto di Matematica dell'Universita di Trieste
In this note we continue the investigation carried out in [Topology Appl. 65, No. 1, 101-104 (199... more In this note we continue the investigation carried out in [Topology Appl. 65, No. 1, 101-104 (1995; Zbl 0828.54024)]. We show that every weighted quasi-metric space can be identified with a subspace of a space of some canonical type, which is constructed from a metric space. We also present a very simple method to construct a weighted quasi-metric space, as the graph of a function defined on a metric space, and show that every weighted quasi-metric space arises in this way. Similar results may be obtained if we drop the requirement that the weight function have nonnegative values.
We show that every quasi-metric space is isomorphic to a subspace of the hyperspace of a suitable... more We show that every quasi-metric space is isomorphic to a subspace of the hyperspace of a suitable metric space, endowed with the Hausdorff quasi-metric. Therefore a topological space is quasi-metrizable if and only if it can be embedded in a Hausdorff quasi-metric hyperspace.
A structure called d 0 -algebra is defined and studied, which leads to a new description of D-lat... more A structure called d 0 -algebra is defined and studied, which leads to a new description of D-lattices and generalized D-lattices by means of only one totally defined operation. c 2011 Mathematical Institute Slovak Academy of Sciences 2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 03G05; Secondary 03G25, 81P10. K e y w o r d s: set-difference, Boolean algebra, D-lattice, D-poset, generalized D-lattice, congruence, ideal.
We prove an algebraic and a topological decomposition theorem for complete D-lattices (i.e., latt... more We prove an algebraic and a topological decomposition theorem for complete D-lattices (i.e., lattice-ordered effect algebras). As a consequence, we obtain a Hammer-Sobczyk type decomposition theorem for modular measures on D-lattices.
ABSTRACT Bornological convergence is a generalization of the well known Attouch--Wets convergence... more ABSTRACT Bornological convergence is a generalization of the well known Attouch--Wets convergence. Our aim is to compare lower, upper and ``two-sided'' convergences generated by two compatible uniformities and two arbitrary bornologies. Moreover, the comparison of convergences induced by bounded-proximal topologies is characterized.
Given a Hausdorff space X, we calculate the tightness and the character of the hyperspace CL ∅ (X... more Given a Hausdorff space X, we calculate the tightness and the character of the hyperspace CL ∅ (X) of X, endowed with either the co-compact or the lower Vietoris topology, and give some estimates for the tightness of CL ∅ (X), endowed with the Fell topology.
Let L be a D-lattice, i.e. a lattice ordered effect algebra, and let BV be the Banach space of al... more Let L be a D-lattice, i.e. a lattice ordered effect algebra, and let BV be the Banach space of all real-valued functions of bounded variation on L (vanishing at 0) endowed with the variation norm. We prove the existence of a continuous Aumann-Shapley value ϕ on bv NA, the subspace of BV spanned by all functions of the form f • µ, where µ : L → [0, 1] is a nonatomic σ-additive modular measure and f : [0, 1] → R is of bounded variation and continuous at 0 and at 1. Mathematics Subject Classification (2000). 28A12, 06C15, 91A12. A. Basile et al. Positivity (D1) a b is defined if and only if b ≤ a; (D2) If b = b c. Vol. 12 (2008) On the Aumann-Shapley value 615
In 1986, A. Basile and H. Weber proved that every countable family M of σ-additive measures on a ... more In 1986, A. Basile and H. Weber proved that every countable family M of σ-additive measures on a σcomplete Boolean ring R admits a dense G δ -set of separating points, with respect to a suitable topology, in the following two cases: either (i) each element of M is s-bounded and, whenever λ, ν ∈ M are distinct, the quotient of R modulo N (λ − ν) is infinite, or (ii) each element of M is continuous.
Beer and Di Concilio have given necessary and sufficient conditions for a two-sided Attouch--Wets... more Beer and Di Concilio have given necessary and sufficient conditions for a two-sided Attouch--Wets topology to contain another on the hyperspace of non-empty closed subsets of a metrizable space as determined by metrics compatible with the topology. In the present paper, we characterize comparability of lower Attouch--Wets topologies as determined by compatible metrics.
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