Applied General Topology - Vol 09, No 2 (2008)

Tabla de contenidos

Semigroups and their topologies arising from Green's left quasiorder
Spread of balleans
On pseudo-k-spaces
A note on a fixed point theorem for ray oriented maps
Asymptotic proximities
Unitary representability of free abelian topological groups
On the topology of generalized quotients
Applications of pre-open sets
Probability measure monad on the category of ultrametric spaces
Generalizations of Z-supercontinuous functions and Dδ-supercontinuous functions
The Jordan curve theorem in the Khalimsky plane
On the continuity of factorizations
Scott-representability of some spaces of Tall and Miskin
On σ-starcompact spaces
On some variations of multifunction continuity
Review of the book of Vladimir Kovalevsky "Geometry of Locally Finite Spaces"

URI permanente para esta colecciónhttps://riunet.upv.es/handle/10251/86405

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Access status: Abierto ,
Review of the book of Vladimir Kovalevsky "Geometry of Locally Finite Spaces"
(Universitat Politècnica de València, 2008-10-01) Matveev, Sergei
No abstract.
Access status: Abierto ,
On some variations of multifunction continuity
(Universitat Politècnica de València, 2008-10-01) Kanibir, A.; Reilly, Ivan L.
[EN] This paper considers six classes of multifunctions between topological spaces, namely almost ℓ−continuous multifunctions, K-almost c−continuous multifunctions, nearly continuous multifunctions, almost nearly continuous multifunctions, super continuous multifunctions, and δ−continuous multifunctions. We relate these classes of multifunctions to others, and provide characterizations of related concepts especially in terms of appropriate changes of topology.
Access status: Abierto ,
On σ-starcompact spaces
(Universitat Politècnica de València, 2008-10-01) Song, Yan-Kui; National Science Foundation, China
[EN] A space X is σ-starcompact if for every open cover U of X, there exists a σ-compact subset C of X such that St(C, U) = X. We investigate the relations between σ-starcompact spaces and other related spaces, and also study topological properties of σ-starcompact spaces.
Access status: Abierto ,
Scott-representability of some spaces of Tall and Miskin
(Universitat Politècnica de València, 2008-10-01) Bennet, Harold; Lutzer, David
[EN] In this paper we show that a variation of a technique of Miskin and Tall yields a cocompact completely regular Moore space that is Scott-domain-representable and has a closed Gδ-subspace that is not Scott-domain-representable. This clarifies the general topology of Scott-domain-representable spaces and raises additional questions about Scott-domain representability in Moore spaces.
Access status: Abierto ,
On the continuity of factorizations
(Universitat Politècnica de València, 2008-10-01) Comfort, W.W.; Gotchev, Ivan S.; Recoder-Nuñez, Luis
[EN] Let {Xi : i ∈ I} be a set of sets, XJ :=Пi∈J Xi when Ø ≠ J ⊆ I; Y be a subset of XI , Z be a set, and f : Y → Z. Then f is said to depend on J if p, q ∈ Y , pJ = qJ ⇒ f(p) = f(q); in this case, fJ : πJ [Y ] → Z is well-defined by the rule f = fJ ◦ πJ|Y When the Xi and Z are spaces and f : Y → Z is continuous with Y dense in XI , several natural questions arise: (a) does f depend on some small J ⊆ I? (b) if it does, when is fJ continuous? (c) if fJ is continuous, when does it extend to continuous fJ : XJ → Z? (d) if fJ so extends, when does f extend to continuous f : XI → Z? (e) if f depends on some J ⊆ I and f extends to continuous f : XI → Z, when does f also depend on J? The authors offer answers (some complete, some partial) to some of these questions, together with relevant counterexamples. Theorem 1. f has a continuous extension f : XI → Z that depends on J if and only if fJ is continuous and has a continuous extension fJ : XJ → Z. Example 1. For ω ≤ k ≤ c there are a dense subset Y of [0, 1]k and f ∈ C(Y, [0, 1]) such that f depends on every nonempty J ⊆ k, there is no J ∈ [k]<ω such that fJ is continuous, and f extends continuously over [0, 1]k. Example 2. There are a Tychonoff space XI, dense Y ⊆ XI, f ∈ C(Y ), and J ∈ [I]<ω such that f depends on J, πJ [Y ] is C-embedded in XJ , and f does not extend continuously over XI .
Access status: Abierto ,
The Jordan curve theorem in the Khalimsky plane
(Universitat Politècnica de València, 2008-10-01) Bouassida, Ezzeddine
[EN] The connectivity in Alexandroff topological spaces is equivalent to the path connectivity. This fact gets some specific properties to Z2, equipped with the Khalimsky topology. This allows a sufficiently precise description of the curves in Z2 and permit to prove a digital Jordan curve theorem in Z2.
Access status: Abierto ,
Generalizations of Z-supercontinuous functions and Dδ-supercontinuous functions
(Universitat Politècnica de València, 2008-10-01) Kohli, J.K.; Singh, D.; Kumar, Rajesh
[EN] Two new classes of functions, called ‘almost z-supercontinuous functions’ and ’almost Dδ-supercontinuous functions’ are introduced. The class of almost z-supercontinuous functions properly includes the class of z-supercontinuous functions (Indian J. Pure Appl. Math. 33(7), (2002), 1097-1108) as well as the class of almost clopen maps due to Ekici (Acta. Math. Hungar. 107(3), (2005), 193-206) and is properly contained in the class of almost Dδ-supercontinuous functions which in turn constitutes a proper subclass of the class of almost strongly θ-continuous functions due to Noiri and Kang (Indian J. Pure Appl. Math. 15(1), (1984), 1-8) and which in its turn include all δ-continuous functions of Noiri (J. Korean Math. Soc. 16 (1980), 161-166). Characterizations and basic properties of almost z-supercontinuous functions and almost Dδ-supercontinuous functions are discussed and their place in the hierarchy of variants of continuity is elaborated. Moreover, properties of almost strongly θ-continuous functions are investigated and sufficient conditions for almost strongly θ-continuous functions to have u θ-closed (θ-closed) graph are formulated.
Access status: Abierto ,
Probability measure monad on the category of ultrametric spaces
(Universitat Politècnica de València, 2008-10-01) Hubal, O.B.; Zarichnyi, M.M.
[EN] The set of all probability measures with compact support on an ultrametric space can be endowed with a natural ultrametric. We show that the functor of probability measures with finite supports (respectively compact supports) forms a monad in the category of ultrametric spaces (respectively complete ultrametric spaces) and nonexpanding maps. It is also proven that the G-symmetric power functor has an extension onto the Kleisli category of the probability measure monad.
Access status: Abierto ,
Applications of pre-open sets
(Universitat Politècnica de València, 2008-10-01) Jun, Young Bae; Jeong, Seong Woo; Lee, Hyeon Jeong; Lee, Joon Woo
[EN] Using the concept of pre-open set, we introduce and study topological properties of pre-limit points, pre-derived sets, preinterior and pre-closure of a set, pre-interior points, pre-border, prefrontier and pre-exterior. The relations between pre-derived set (resp. pre-limit point, pre-interior (point), pre-border, pre-frontier, and preexterior) and α-derived set (resp. α-limit point, α-interior (point), α-border, α-frontier, and α-exterior) are investigated
Access status: Abierto ,
On the topology of generalized quotients
(Universitat Politècnica de València, 2008-10-01) Burzyk, Józef; Ferens, Cezary; Mikusinski, Piotr
[EN] Generalized quotients are defined as equivalence classes of pairs (x, f), where x is an element of a nonempty set X and f is an element of a commutative semigroup G acting on X. Topologies on X and G induce a natural topology on B(X,G), the space of generalized quotients. Separation properties of this topology are investigated.
Access status: Abierto ,
Unitary representability of free abelian topological groups
(Universitat Politècnica de València, 2008-10-01) Uspenskij, Vladimir V.
[EN] For every Tikhonov space X the free abelian topological group A(X) and the free locally convex vector space L(X) admit a topologically faithful unitary representation. For compact spaces X this is due to Jorge Galindo.
Access status: Abierto ,
Asymptotic proximities
(Universitat Politècnica de València, 2008-10-01) Protasov, Igor V.
[EN] A ballean is a set endowed with some family of subsets which are called the balls. The properties of the family of balls are postulated in such a way that the balleans can be considered as a natural asymptotic counterparts of the uniform topological spaces. We introduce and study an asymptotic proximity as a counterpart of proximity relation for uniform topological space.
Access status: Abierto ,
A note on a fixed point theorem for ray oriented maps
(Universitat Politècnica de València, 2008-10-01) Arockiasamy, A.; Eldred, Anthony
[EN] In this paper, we will prove a fixed point theorem for a ray-oriented map defined on a nonempty closed bounded convex subset of a Banach space.
Access status: Abierto ,
On pseudo-k-spaces
(Universitat Politècnica de València, 2008-10-01) Miranda, Anna Maria
[EN] In this note a new class of topological spaces generalizingk-spaces, the pseudo-k-spaces, is introduced and investigated. Particu-lar attention is given to the study of products of such spaces, in analogyto what is already known aboutk-spaces and quasi-k-spaces
Access status: Abierto ,
Spread of balleans
(Universitat Politècnica de València, 2008-10-01) Filali, Mahmoud; Protasov, Igor V.
[EN] A ballean is a set endowed with some family of balls in such a way that a ballean can be considered as an asymptotic counterpart of a uniform topological space. We introduce and study a new cardinal invariant called the spread of a ballean. In particular, we show that, for every ordinal ballean B, spread of B coincides with density of B.
Access status: Abierto ,
Semigroups and their topologies arising from Green's left quasiorder
(Universitat Politècnica de València, 2008-10-01) Richmond, Bettina
[EN] Given a semigroup (S, ·), Green’s left quasiorder on S is given by a ≤ b if a = u · b for some u ϵ S1. We determine which topological spaces with five or fewer elements arise as the specialization topology from Green’s left quasiorder for an appropriate semigroup structure on the set. In the process, we exhibit semigroup structures that yield general classes of finite topological spaces, as well as general classes of topological spaces which cannot be derived from semigroup structures via Green’s left quasiorder.

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