Applied General Topology

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Applied General Topology publica artículos de investigación originales relacionados con las interacciones entre la topología general y otras disciplinas matemáticas así como los resultados topológicos con aplicaciones a otras áreas de la ciencia, y el desarrollo de teorías topológicas de importancia para permitir futuras aplicaciones.

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A better framework for first countable spaces
(Universitat Politècnica de València, 2003-10-01) Preuss, Gerhard
[EN] In the realm of semiuniform convergence spaces first countability is divisible and leads to a well-behaved topological construct with natural function spaces and one-point extensions such that countable products of quotients are quotients. Every semiuniform convergence space (e.g. symmetric topological space, uniform space, filter space, etc.) has an underlying first countable space. Several applications of first countability in a broader context than the usual one of topological spaces are studied.
A class of ideals in intermediate rings of continuous functions
(Universitat Politècnica de València, 2019-04-01) Bag, Sagarmoy; Acharyya, Sudip Kumar; Mandal, Dhananjoy
[EN] For any completely regular Hausdorff topological space X, an intermediate ring A(X) of continuous functions stands for any ring lying between C∗(X) and C(X). It is a rather recently established fact that if A(X) ≠ C(X), then there exist non maximal prime ideals in A(X).We offer an alternative proof of it on using the notion of z◦-ideals. It is realized that a P-space X is discrete if and only if C(X) is identical to the ring of real valued measurable functions defined on the σ-algebra β(X) of all Borel sets in X. Interrelation between z-ideals, z◦-ideal and ƷA-ideals in A(X) are examined. It is proved that within the family of almost P-spaces X, each ƷA -ideal in A(X) is a z◦-ideal if and only if each z-ideal in A(X) is a z◦-ideal if and only if A(X) = C(X).
A class of spaces containing all generalized absolutely closed (almost compact) spaces
(Universitat Politècnica de València, 2006-10-01) Kohli, J.K.; Das, A.K.
[EN] The class of θ-compact spaces is introduced which properly contains the class of almost compact (generalized absolutely closed) spaces and is strictly contained in the class of quasicompact spaces. In the realm of almost regular spaces, the class of θ-compact spaces coincides with the class of nearly compact spaces. Moreover, an almost regular θ-compact space is mildly normal (= K-normal). A θ-closed, θ-embedded subset of a θ-compact space is θ-compact and the product of two θ-compact space is θ-compact if one of them is compact. A (strongly) θ-continuous image of a θ-compact space is θ-compact (compact). A space is compact if and only if it is θ-compact and θ-point paracompact.
A construction of a fuzzy topology from a strong fuzzy metric
(Universitat Politècnica de València, 2016-10-03) Grecova, Svetlana; Sostak, Alexander; Uljane, Ingrida
[EN] After the inception of the concept of a fuzzy metric by I. Kramosil and J. Michalek, and especially after its revision by A. George and G. Veeramani, the attention of many researches was attracted to the topology induced by a fuzzy metric. In most of the works devoted to this subject the resulting topology is an ordinary, that is a crisp one. Recently some researchers showed interest in the fuzzy-type topologies induced by fuzzy metrics. In particular, in the paper (J.J. Mi\~{n}ana, A. \v{S}ostak, {\it Fuzzifying topology induced by a strong fuzzy metric}, Fuzzy Sets and Systems, 6938 DOI information: 10.1016/j.fss.2015.11.005.) a fuzzifying topology ${\mathcal T}:2^X \to [0,1]$ induced by a fuzzy metric $m: X\times X \times [0,\infty)$ was constructed. In this paper we extend this construction to get the fuzzy topology ${\mathcal T}: [0,1]^X \to [0,1]$ and study some properties of this fuzzy topology.54A
A contribution to fuzzy subspaces
(Universitat Politècnica de València, 2002-04-01) Alamar, Miguel; Estruch, Vicente D.; Universitat Politècnica de València
[EN] We give a new concept of fuzzy topological subspace, which extends the usual one, and study in it the related concepts of interior, closure and conectedness.
A contribution to the study of fuzzy metric spaces
(Universitat Politècnica de València, 2001-04-01) Sapena Piera, Almanzor; Dpto. de Matemática Aplicada; Escuela Politécnica Superior de Gandia; Instituto de Investigación para la Gestión Integrada de Zonas Costeras
[EN] We give some examples nad properties of fuzzy metric spaces, in the sense of George and Veramani, and characterize the To topological spaces which admit a compatible uniformity that has a countable transitive base, in terms of the fuzzy theory.
A countably compact free Abelian group whose size has countable cofinality
(Universitat Politècnica de València, 2004-04-01) Castro Pereira, I.; Tomita, A.H.; Ministry of Education, Culture, Sports, Science and Technology, Japón
[EN] Based on some set-theoretical observations, compactness results are given for general hit-and-miss hyperspaces. Compactness here is sometimes viewed splitting into “k-Lindelöfness” and ”k-compactness” for cardinals k. To focus only hit-and-miss structures, could look quite old-fashioned, but some importance, at least for the techniques, is given by a recent result of Som Naimpally, to who this article is hearty dedicated.
A counter example on a Borsuk conjecture
(Universitat Politècnica de València, 2023-04-05) Cholaquidis, Alejandro
[EN] The study of shape restrictions of subsets of Rd has several applications in many areas, being convexity, r-convexity, and positive reach, some of the most famous, and typically imposed in set estimation. The following problem was attributed to K. Borsuk, by J. Perkal in 1956: find an r-convex set which is not locally contractible. Stated in that way is trivial to find such a set. However, if we ask the set to be equal to the closure of its interior (a condition fulfilled for instance if the set is the support of a probability distribution absolutely continuous with respect to the d-dimensional Lebesgue measure), the problem is much more difficult. We present a counter example of a not locally contractible set, which is r-convex. This also proves that the class of supports with positive reach of absolutely continuous distributions includes strictly the class ofr-convex supports of absolutely continuous distributions.
A curious example involving ordered compactifications
(Universitat Politècnica de València, 2002-10-01) Richmond, Thomas A.
[EN] For a certain product X x Y where X is compact, connected, totally ordered space, we find that the semilattice K0 (X x Y) of ordered compactifications of X x Y is isomorphic to a collection of Galois connections and to a collection of functions F which determines a quasi-uniformity on an extended set X U {+∞}, from which the topology and order on X is easily recovered. It is well-known that each ordered compactification of an ordered space X x Y corresponds to a totally bounded quasi-uniformity on X x Y compatible with the topology and order on X x Y, and thus K0 (X x Y) may be viewed as a collection of quasi-uniformities on X x Y. By the results here, these quasi-uniformities on X x Y determine a quasi-uniformity on the related space X U {+∞}.
A decomposition of normality via a generalization of $\kappa$-normality
(Universitat Politècnica de València, 2017-10-02) Das, Ananga Kumar; Bhat, Pratibha
[EN] A simultaneous generalization of $\kappa$-normality and weak $\theta$-normality is introduced. Interrelation of this generalization of normality with existing variants of normality is studied.In the process of investigation a new decomposition of normality is obtained.
A disjointly tight irresolvable space
(Universitat Politècnica de València, 2020-10-01) Bella, Angelo; Hrusak, Michael; Università di Catania; Universidad Nacional Autónoma de México; Consejo Nacional de Ciencia y Tecnología, México
[EN] In this short note we prove the existence (in ZFC) of a completely regular countable disjointly tight irresolvable space by showing that every sub-maximal countable dense subset of 2c is disjointly tight.
A fuzzification of the category of M-valued L-topological spaces
(Universitat Politècnica de València, 2004-10-01) Kubiak, Tomasz; Sostak, Alexander P.
[EN] A fuzzy category is a certain superstructure over an ordinary category in which ”potential” objects and ”potential” morphisms could be such to a certain degree. The aim of this paper is to introduce a fuzzy category FTOP(L,M) extending the category TOP(L,M) of M-valued L- topological spaces which in its turn is an extension of the category TOP(L) of L-fuzzy topological spaces in Kubiak-Sostak’s sense. Basic properties of the fuzzy category FTOP(L,M) and its objects are studied.
A generalized coincidence point index
(Universitat Politècnica de València, 2005-04-01) Benkafadar, N.M.; Benkara-Mostefa, M.C.; Agence Nationale pour le Développement de la Recherche Universitaire, Argelia; Ministère de l'Enseignement Supérieur et de la Recherche Scientifique, Argelia
[EN] The paper is devoted to build for some pairs of continuous single-valued maps a coincidence point index. The class of pairs (f, g) satisfies the condition that f induces an epimorphism of the Cech homology groups with compact supports and coefficients in the field of rational numbers Q. Using this concept one defines for a class of multi-valued mappings a fixed point degree. The main theorem states that if the general coincidence point index is different from {0}, then the pair (f, g) admits at least a coincidence point. The results may be considered as a generalization of the above Eilenberg-Montgomery theorems [12], they include also, known fixed-point and coincidence-point theorems for single-valued maps and multi-valued transformations.
A generalized version of the rings CK (X) and C∞(X)– an enquery about when they become Noetheri
(Editorial Universitat Politècnica de València, 2015-04-01) Acharyya, Sudip Kumar; Chattopadhyay, Kshitish Chandra; Rooj, Pritam; University Grants Commission, India
[EN] Suppose F is a totally ordered field equipped with its order topology and X a completely F-regular topological space. Suppose P is an ideal of closed sets in X and X is locally-P. Let CP(X, F) = {f : X ! F | f is continuous on X and its support belongs to P} and CP∞ (X, F) = {f 2 CP(X, F) | 8" > 0 in F, clX{x 2 X : |f(x)| > "} 2 P}. Then CP(X, F) is a Noetherian ring if and only if CP∞ (X, F) is a Noetherian ring if and only if X is a finite set. The fact that a locally compact Hausdorff space X is finite if and only if the ring CK(X) is Noetherian if and only if the ring C∞(X) is Noetherian, follows as a particular case on choosing F = R and P = the ideal of all compact sets in X. On the other hand if one takes F = R and P = the ideal of closed relatively pseudocompact subsets of X, then it follows that a locally pseudocompact space X is finite if and only if the ring C (X) of all real valued continuous functions on X with pseudocompact support is Noetherian if and only if the ring C ∞ (X) = {f 2 C(X) | 8" > 0, clX{x 2 X : |f(x)| > "} is pseudocompact } is Noetherian. Finally on choosing F = R and P = the ideal of all closed sets in X, it follows that: X is finite if and only if the ring C(X) is Noetherian if and only if the ring C∗(X) is Noetherian.
A glance into the anatomy of monotonic maps
(Universitat Politècnica de València, 2021-04-01) Buzyakova, Raushan
[EN] Given an autohomeomorphism on an ordered topological space or its subspace, we show that it is sometimes possible to introduce a new topology-compatible order on that space so that the same map is monotonic with respect to the new ordering. We note that the existence of such a re-ordering for a given map is equivalent to the map being conjugate (topologically equivalent) to a monotonic map on some homeomorphic ordered space. We observe that the latter cannot always be chosen to be order-isomorphic to the original space. Also, we identify other routes that may lead to similar affirmative statements for other classes of spaces and maps.
A Kuratowski-Mrówka type characterization of fibrewise compactness
(Universitat Politècnica de València, 2011-04-01) Neira U.; Clara M.
[EN] In this paper a Kuratowski-Mrówka type characterization of fibrewisecompact topological spaces is presented.
A local fixed point theorem for set-valued mappings on partial metric spaces
(Universitat Politècnica de València, 2016-04-12) Benterki, Abdessalem
[EN] The purpose of this paper is to study the existence and location of fixed points for pseudo-contractive-type set-valued mappings in the setting of partial metric spaces by using Bianchini-Grundolfi gauge functions.
A new cardinal function on topological spaces
(Universitat Politècnica de València, 2017-04-03) Sari, Dewi Kartika; Zhao, Dongsheng; Ministry of Finance, Indonesia
[EN] Using neighbourhood assignments, we introduce and study a new cardinal function, namely GCI(X), for every topological space X. We shall mainly investigate the spaces X with finite GCI(X). Some properties of this cardinal in connection with special types of mappings are also proved. Guardar / Salir Siguiente >
A new topology over the primary-like spectrum of a module
(Universitat Politècnica de València, 2021-10-01) Rashedi, Fatemeh
[EN] Let R be a commutative ring with identity and M a unitary R-module. The primary-like spectrum SpecL(M) is the collection of all primary-like submodules Q of M, the recent generalization of primary ideals, such that M/Q is a primeful R-module. In this article, we topologies SpecL(M) with the patch-like topology, and show that when, SpecL(M) with the patch-like topology is a quasi-compact, Hausdorff, totally disconnected space.
A non-discrete space X with Cp(X) Menger at infinity
(Universitat Politècnica de València, 2019-04-01) Bella, Angelo; Hernández-Gutiérrez, Rodrigo; Secretaría de Educación Pública, México
[EN] In a paper by Bella, Tokgös and Zdomskyy it is asked whether there exists a Tychonoff space X such that the remainder of Cp(X) in some compactification is Menger but not σ-compact. In this paper we prove that it is consistent that such space exists and in particular its existence follows from the existence of a Menger ultrafilter.

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