46 results on '"Compact space"'
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2. σ-Prime Spectrum of Almost Distributive Lattices.
- Author
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Noorbhasha, Rafi, Bandaru, Ravikumar, and Iampan, Aiyared
- Subjects
- *
PRIME ideals , *COMPACT spaces (Topology) , *DISTRIBUTIVE lattices - Abstract
For each α-ideal of an almost distributive lattice (ADL) to become a σ-ideal, a set of equivalent conditions is derived, which tends to result in a characterization of generalized Stone ADLs. On an ADL, a one-to-one correspondence is derived between the set of all prime σ-ideals of the ADL and the set of all prime σ-ideals of the quotient ADL. Finally, proved some properties of prime σ-ideals of a normal ADL topologically. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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3. COMPACTNESS AND CARDINALITY OF THE SPACE OF CONTINUOUS FUNCTIONS UNDER REGULAR TOPOLOGY.
- Author
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Aaliya, Mir and Mishra, Sanjay
- Subjects
FUNCTION spaces ,CONTINUOUS functions ,TOPOLOGY ,REGULAR graphs ,COMPACT spaces (Topology) ,TOPOLOGICAL spaces - Abstract
In this paper, we investigate the compactness and cardinality of the space C(X, Y ) of continuous functions from a topological space X to Y equipped with the regular topology. We prove that different forms of compactness, such as sequential compactness, countable compactness, and pseudocompactness, coincide on a subset of C(X, Y ) with regular topology. Moreover, we prove the comparison and coincidence of regular topology with the graph topology on the space C(X, Y ). Furthermore, we examine various cardinal invariants, such as density, character, pseudocharacter, etc., on the space C(X, Y ) equipped with the regular topology. In addition, we define a type of equivalence between X and Y in terms of C(X) and C(Y ) endowed with the regular topology and investigate certain cardinal invariants preserved by this equivalence. [ABSTRACT FROM AUTHOR]
- Published
- 2024
4. Smallness in topology.
- Author
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Adámek, Jiří, Hušek, Miroslav, Rosický, Jiří, and Tholen, Walter
- Subjects
TOPOLOGICAL spaces ,TOPOLOGY ,HAUSDORFF spaces ,ALGEBRA ,COMPACT spaces (Topology) ,ABELIAN categories ,HOMOTOPY theory - Abstract
Quillen's notion of small object and the Gabriel-Ulmer notion of finitely presentable or generated object are fundamental in homotopy theory and categorical algebra. Do these notions always lead to rather uninteresting classes of objects in categories of topological spaces, such as all finite discrete spaces, or just the empty space, as the examples and remarks in the existing literature may suggest? This article demonstrates that the establishment of full characterizations of these notions (and some natural variations thereof) in many familiar categories of spaces can be quite challenging and may lead to unexpected surprises. In fact, we show that there are significant differences in this regard even amongst the categories defined by the standard separation axioms, with the T
1 -separation condition standing out. The findings about these specific categories lead us to insights also when considering rather arbitrary full reflective subcategories of the category of all topological spaces. [ABSTRACT FROM AUTHOR]- Published
- 2023
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5. Linear equivalence of (pseudo) compact spaces.
- Author
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Baars, Jan, van Mill, Jan, and Tkachuk, Vladimir V.
- Subjects
COMPACT spaces (Topology) ,COMMERCIAL space ventures ,METRIC spaces ,FUNCTION spaces - Abstract
Given Tychonoff spaces X and Y, Uspenskij proved in [15] that if X is pseudocompact and Cp(X) is uniformly homeomorphic to C
p (Y), then Y is also pseudocompact. In particular, if Cp (X) is linearly homeomorphic to Cp (Y), then X is pseudocompact if and only if so is Y. This easily implies Arhangel'skii's theorem [1] which states that, in the case when Cp (X) is linearly homeomorphic to Cp (Y the space X is compact if and only if Y is compact. We will establish that existence of a linear homeomorphism between the spaces Cp *(X) and Cp *(Y) implies that X is (pseudo)compact if and only if so is Y. We will also show that the methods of proof used by Arhangel'skii and Uspenskij do not work in our case. [ABSTRACT FROM AUTHOR]- Published
- 2023
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6. Topological Entropy for Arbitrary Subsets of Infinite Product Spaces.
- Author
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Sadr, Maysam Maysami and Shahrestani, Mina
- Subjects
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TOPOLOGICAL entropy , *METRIC spaces , *TOPOLOGICAL spaces , *SEQUENCE spaces , *ORBITS (Astronomy) , *COMPACT spaces (Topology) , *INFINITE processes , *ENTROPY - Abstract
In this note, a notion of generalized topological entropy for arbitrary subsets of the space of all sequences in a compact topological space is introduced. It is shown that for a continuous map on a compact space, the generalized topological entropy of the set of all orbits of the map coincides with the classical topological entropy of the map. Some basic properties of this new notion of entropy are considered; among them are the behavior of the entropy with respect to disjoint union, cartesian product, component restriction and dilation, shift mapping, and some continuity properties with respect to Vietoris topology. As an example, it is shown that any self-similar structure of a fractal given by a finite family of contractions gives rise to a notion of intrinsic topological entropy for subsets of the fractal. A generalized notion of Bowen's entropy associated to any increasing sequence of compatible semimetrics on a topological space is introduced and some of its basic properties are considered. As a special case for 1 ≤ p ≤ ∞ , the Bowen p-entropy of sets of sequences of any metric space is introduced. It is shown that the notions of generalized topological entropy and Bowen ∞ -entropy for compact metric spaces coincide. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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7. On Condensations onto σ-Compact Spaces.
- Author
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Lipin, A. E. and Osipov, A. V.
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METRIC spaces , *CARDINAL numbers , *CONDENSATION , *COMPACT spaces (Topology) , *BIJECTIONS - Abstract
In this paper, we prove the following result. Let X be a complete metric space of weight and be a set such that . Then there is no continuous bijection of the subspace onto a -compact space. As a result, there is no continuous bijection of the subspace onto a Polish space. Thus, it has been proved that metric compact spaces are not -spaces for any uncountable cardinal number . This result answers the question asked by E.G. Pytkeev in his coauthored work "On the properties of subclasses of weakly dyadic compact sets" to be published in the Siberian Mathematical Journal. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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8. Space of Stone-Čech Compactification 𝜷ℕ.
- Author
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Ridha, Haider Mohammed and Al-Fayadh, Ali Hassan Nasser
- Subjects
TOPOLOGICAL property ,NATURAL numbers ,COMPACT spaces (Topology) - Abstract
Copyright of Diyala Journal for Pure Science is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
- Published
- 2022
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9. Functional countability is preserved by some products.
- Author
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Tkachuk, V. V.
- Subjects
- *
COMPACT spaces (Topology) , *COMMERCIAL space ventures - Abstract
Given a functionally countable space X, if K is a scattered Corson compact space, then the product X × K is functionally countable. If the space X is, additionally, weakly ω 1 -Lindelöf, then X × L is functionally countable for any scattered Lindelöf space L. In particular, if X is a weakly Lindelöf functionally countable space, then the product X × L is functionally countable whenever L is a scattered Lindelöf space. We also establish that any finite product of countably compact functionally countable spaces is functionally countable and exponential separability of countably compact spaces X and Y implies that X × Y is exponentially separable if one of them is sequential. If the product X = X 1 × ⋯ × X n of functionally countable spaces X 1 ,... , X n is pseudocompact, then X is functionally countable. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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10. AMENDMENT TO "LINDELÖF WITH RESPECT TO AN IDEAL" [NEW ZEALAND J. MATH. 42, 115-120, 2012.
- Author
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HOQUE, JIARUL and MODAK, SHYAMAPADA
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MATHEMATICS , *COMPACT spaces (Topology) - Abstract
We give a counterexample in this amendment to show that there is an error in consideration of the statement "if f: X → Y and J is an ideal on Y, then f-1(J) = ff-1(J): J ℇ J- is an ideal on X" by Hamlett in his paper "Lindelöf with respect to an ideal" [New Zealand J. Math. 42, 115-120, 2012]. We also modify it here in a new way and henceforth put forward correctly all the results that were based on the said statement derived therein. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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11. A generalized Powers averaging property for commutative crossed products.
- Author
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Amrutam, Tattwamasi and Ursu, Dan
- Subjects
- *
HAUSDORFF spaces , *GENERALIZED spaces , *HOMEOMORPHISMS , *C*-algebras , *COMPACT spaces (Topology) , *SIMPLICITY - Abstract
We prove a generalized version of Powers' averaging property that characterizes simplicity of reduced crossed products C(X) \rtimes _\lambda G, where G is a countable discrete group, and X is a compact Hausdorff space which G acts on minimally by homeomorphisms. As a consequence, we generalize results of Hartman and Kalantar on unique stationarity to the state space of C(X) \rtimes _\lambda G and to Kawabe's generalized space of amenable subgroups \operatorname {Sub}_a(X,G). This further lets us generalize a result of the first named author and Kalantar on simplicity of intermediate C*-algebras. We prove that if C(Y) \subseteq C(X) is an inclusion of unital commutative G-C*-algebras with X minimal and C(Y) \rtimes _\lambda G simple, then any intermediate C*-algebra A satisfying C(Y) \rtimes _\lambda G \subseteq A \subseteq C(X) \rtimes _\lambda G is simple. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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12. Some classes of topological spaces related to zero-sets.
- Author
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GOLRIZKHATAMI, F. and TAHERIFAR, A.
- Subjects
- *
TOPOLOGICAL property , *TOPOLOGICAL spaces , *COMPACT spaces (Topology) - Abstract
An almost P-space is a topological space in which every zero-set is regular-closed. We introduce a large class of spaces, C-almost P-space (briey CAP-space), consisting of those spaces in which the closure of the interior of every zero-set is a zero-set. In this paper we study CAP-spaces. It is proved that if X is a dense and Z#-embedded sub-space of a space T, then T is CAP if and only if X is a CAP and CRZ-extended in T (i.e, for each regular-closed zero-set Z in X, clT Z is a zero-set in T). In 6P.5 of [8] it was shown that a closed count- able union of zero-sets need not be a zero-set. We call X a CZ-space whenever the closure of any countable union of zero-sets is a zero-set. This class of spaces contains the class of P-spaces, perfectly normal spaces, and is contained in the cozero complemented spaces and CAP-spaces. In this paper we study topological properties of CZ (resp. coz-ero complemented)-space and other classes of topological spaces near to them. Some algebraic and topological equivalent conditions of CZ (resp. cozero complemented)-space are characterized. Examples are provided to illustrate and delimit our results. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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13. Fuzzy Structure Space of Semigroups.
- Author
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Mandal, Manasi and (Goswami), Sarbani Mukherjee
- Subjects
- *
PRIME ideals , *AXIOMS , *COMPACT spaces (Topology) - Abstract
In this paper the fuzzy structure space of a semigroup has been introduced. Among other things, the separation axioms and compactness in the fuzzy structure space of a semigroup have been studied. [ABSTRACT FROM AUTHOR]
- Published
- 2020
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14. On cellular-compact spaces.
- Author
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Juhász, I., Soukup, L., and Szentmiklóssy, Z.
- Subjects
- *
COMPACT spaces (Topology) , *MERGERS & acquisitions , *TOPOLOGICAL spaces , *SPACE - Abstract
As it was introduced by Tkachuk and Wilson in [7], a topological space X is cellular-compact if for any cellular, i.e. disjoint, family U of non-empty open subsets of X there is a compact subspace K ⊂ X such that K ∩ U ≠ ∅ for each U ∈ U . In this note we answer several questions raised in [7] by showing that any first countable cellular-compact T2-space is T3, and so its cardinality is at most c = 2 ω ; cov (M) > ω 1 implies that every first countable and separable cellular-compactT2-space is compact; if there is no S-space then any cellular-compact T3-space of countable spread is compact; M A ω 1 implies that every point of a compact T2-space of countable spread has a disjoint local π -base. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
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15. CompactNet: learning a compact space for face presentation attack detection.
- Author
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Li, Lei, Xia, Zhaoqiang, Jiang, Xiaoyue, Roli, Fabio, and Feng, Xiaoyi
- Subjects
- *
HUMAN facial recognition software , *COMPACT spaces (Topology) , *COST functions , *VIDEO compression - Abstract
Face presentation attack detection (PAD) has become a clear and present threat for face recognition systems and many countermeasures have been proposed to mitigate it. In these countermeasures, some of them use the features directly extracted from well-known color spaces (e.g., RGB, HSV and YCbCr) to distinguish the fake face images from the genuine ("live") ones. However, the existing color spaces have been originally designed for displaying the visual content of images or videos with high fidelity and are not well suited for directly discriminating the live and fake face images. Therefore, in this paper, we propose a deep-learning system, called CompactNet, for learning a compact space tailored for face PAD. More specifically, the proposed CompactNet does not directly extract the features in existing color spaces, but inputs the color face image into a layer-by-layer progressive space generator. Then, under the optimization of the "points-to-center" triplet loss function, the generator learns a compact space with small intra-class distance, large inter-class distance and a safe interval between different classes. Finally, the feature of the image in compact space is extracted by a pre-trained feature extractor and used for image classification. Reported experiments on three publicly available face PAD databases, namely, the Replay-Attack, the OULU-NPU and the HKBU-MARs V1, show that CompactNet separates very well the two classes of fake and genuine faces and significantly outperforms the state-of-the-art methods for PAD. [ABSTRACT FROM AUTHOR]
- Published
- 2020
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16. Some Remarks on Partial Metric Spaces.
- Author
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Lu, Hanchuan, Zhang, Heng, and He, Wei
- Subjects
- *
COMPACT spaces (Topology) , *METRIC spaces , *TOPOLOGICAL property - Abstract
In this paper, we investigate some topological properties of partial metric spaces (in short PMS). We give some relationship between metric-like PMS, sequentially isosceles PMS and sequentially equilateral PMS. We also prove a type of Urysohn's lemma for metric-like PMS. By applying the construction of Hartman–Mycielski, we show that every bounded PMS can be isometrically embedded into a pathwise connected and locally pathwise connected PMS. In the end, we show that a partial metric space is compact iff it is totally bounded and complete. [ABSTRACT FROM AUTHOR]
- Published
- 2020
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17. LOCALLY ORDERED TOPOLOGICAL SPACES.
- Author
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PIKUL, Piotr
- Subjects
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TOPOLOGICAL spaces , *LINEAR orderings , *TOPOLOGY , *COMPACT spaces (Topology) , *MATHEMATICAL connectedness , *AXIOMS - Abstract
While topology given by a linear order has been extensively studied, this cannot be said about the case when the order is given only locally. The aim of this paper is to fill this gap. We consider relation between local orderability and separation axioms and give characterisation of those regularly locally ordered spaces which are connected, locally connected or Lindelöf. We prove that local orderability is hereditary on open, connected or compact subsets. A collection of interesting examples is also offered. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
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18. Homotopy Properties of the Space If(X) of Idempotent Probability Measures.
- Author
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Zaitov, A. A. and Ishmetov, A. Ya.
- Subjects
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PROBABILITY measures , *COMPACT spaces (Topology) , *SPACE , *HOMOTOPY equivalences - Abstract
A subspace If(X) of the space of idempotent probability measures on a given compact space X is constructed. It is proved that if the initial compact space X is contractible, then If(X) is a contractible compact space as well. It is shown that the shapes of the compact spaces X and If(X) are equal. It is also proved that, given a compact space X, the compact space If(X) is an absolute neighborhood retract if and only if so is X. [ABSTRACT FROM AUTHOR]
- Published
- 2019
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19. d-ideals, f d-ideals and prime ideals.
- Author
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Safaeeyan, S. and Taherifar, A.
- Subjects
PRIME ideals ,ARTIN rings ,COMMUTATIVE rings ,COMPACT spaces (Topology) - Abstract
Let R be a commutative ring. An ideal I of R is called a d-ideal (f d-ideal) provided that for each a ∈ I (finite subset F of I) and b ∈ R, Ann(a) ⊆ Ann(b) (Ann(F) ⊆ Ann(b)) implies that b ∈ I. It is shown that, the class of z
0 -ideals (hence all sz0 -ideals), maximal ideals in an Artinian or in a Kasch ring, annihilator ideals, and minimal prime ideals over a d-ideal are some distinguished classes of d-ideals. Furthermore, we introduce the class of f d-ideals as a subclass of d-ideals in a commutative ring R. In this regard, it is proved that the ring R is a classical ring with property (A) if and only if every maximal ideal of R is an f d-ideal. The necessary and sufficient condition for which every prime f d-ideal of a ring R being a maximal or a minimal prime ideal is given. Moreover, the rings for which their prime d-ideals are z0 -ideals are characterized. Finally, we prove that every prime f d-ideal of a ring R is a minimal prime ideal if and only if for each a ∈ R there exists a finitely generated ideal , for some n ∈ ℕ such that Ann(a, I) = 0. As a consequence, every prime f d-ideal in a reduced ring R is a minimal prime ideal if and only if X= Min(R) is a compact space. [ABSTRACT FROM AUTHOR]- Published
- 2019
- Full Text
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20. On bisequentiality and spaces of strictly decreasing functions on trees.
- Author
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Agostini, Claudio and Somaglia, Jacopo
- Subjects
- *
TREES , *SPACE , *COMPACT spaces (Topology) - Abstract
Abstract We present a characterization of spaces of strictly decreasing functions on trees in terms of bisequentiality. This characterization answers Questions 6.1 and 6.2 of [2]. Moreover we study the relation between these spaces and the classes of Corson, Eberlein and uniform Eberlein compacta. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
21. Fuzzy Structure Space of Semirings and Γ-Semirings.
- Author
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Goswami, Sarbani Mukherjee, Mukhopadhyay, Arup, and Sardar, Sujit Kumar
- Subjects
- *
PRIME ideals , *AXIOMS , *COMPACT spaces (Topology) - Abstract
The purpose of this paper is to study the fuzzy structure space of a semiring as well as of a Γ-semiring. We study separation axioms, compactness etc. in the fuzzy structure space of a semiring. Similar study has also been accomplished in the setting of a Γ-semiring S by using the nice interplay between S and its left operator semiring L. [ABSTRACT FROM AUTHOR]
- Published
- 2019
22. Dependence on ℵ coordinates of separately continuous functions of many variables and its analogs.
- Author
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Mykhaylyuk, Volodymyr
- Subjects
- *
COMMERCIAL space ventures , *COORDINATES , *TOPOLOGICAL spaces , *COMPACT spaces (Topology) - Abstract
We introduce some cardinal functions on the product X 1 × ⋯ × X n of topological spaces X k , which are formulated in terms of the type of local finiteness of families of open sets. Using these cardinal functions, we obtain necessary and sufficient conditions that every separately continuous function or strongly separately continuous function f : X 1 × ⋯ × X n → R depends on ℵ coordinates, where every space X k is a strongly countably Čech complete space. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
23. On a question of Kaplansky.
- Author
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Taherifar, Ali
- Subjects
- *
INTERSECTION theory , *MATHEMATICAL equivalence , *COMPACT spaces (Topology) , *BOREL subsets , *COINCIDENCE theory - Abstract
Kaplansky [7] proved that C K ( X ) is the intersection of all free maximal ideals in C ( X ) in the case of discrete X , and asked whether the equality holds in general. In this paper we prove that C K ( X ) coincides with the intersection of all free maximal ideals if and only if every open hemicompact z -compact (i.e., every zero-set contained in it is compact) subset of X is relatively compact or equivalently, every open Lindelöf z -compact subset of X is relatively compact. We conclude that the equality holds whenever X is a strongly isocompact space. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
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24. If Cp(X) is strongly dominated by a second countable space, then X is countable.
- Author
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Guerrero Sánchez, D. and Tkachuk, V.V.
- Subjects
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COMPACT spaces (Topology) , *DOMINATING set , *IRRATIONAL numbers , *FUNCTION spaces , *CONTINUOUS functions - Abstract
We establish that a Tychonoff space X is countable if and only if C p ( X ) is strongly dominated by a second countable space. The same is true for a compact space K such that C p ( K , [ 0 , 1 ] ) is strongly dominated by a second countable space. We also prove that strong domination by a second countable space of the complement of the diagonal of a Tychonoff space X implies that X is an ℵ 0 -space. Our results solve several published open questions. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
25. Upper Namioka property of compact-valued mappings.
- Author
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Mykhaylyuk, Volodymyr
- Subjects
- *
COMPACT spaces (Topology) , *MATHEMATICAL mappings , *SET theory , *BAIRE spaces , *TOPOLOGICAL spaces - Abstract
We introduce and study the notions of upper Namioka property, upper Namioka space and upper co-Namioka space which are development of the notions of Namioka property, Namioka space and co-Namioka space on the case of compact-valued mappings. We obtain the following results: the class of upper Namioka spaces consists of Baire spaces with everywhere dense set of isolated points; any subset of a upper co-Namioka compact space is separable; every well-ordered upper co-Namioka compact and every upper co-Namioka compact Valdivia are metrizable; the double arrow space is not upper co-Namioka; there exist a compact-valued mapping F ∈ L U ( X , Y ) defined on the product of Namioka and co-Namioka spaces such that F has not upper Namioka property; if there exists a non-metrizable linearly ordered upper co-Namioka space, then the set of its non-isolated neighbor points contains a subset always of the first category subset; every compact-valued mapping F ∈ L U ( X , Y ) defined on the product of a β - σ ′ -unfavorable space X and a separable linearly ordered compact space Y has the upper Namioka property. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
26. Spaces which are retracts or cofactors of paratopological groups.
- Author
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Arhangel'skii, A.V.
- Subjects
- *
TOPOLOGICAL spaces , *BINARY operations , *MATHEMATICS theorems , *COMPACT spaces (Topology) , *THEORY of retracts - Abstract
In this paper we investigate Tychonoff spaces which are retracts of paratopological groups. A strong necessary condition for that is the existence of a certain binary operation on the space (called a τ -twister), which was introduced in [2,3] . Some general theorems are established which imply that βω is not a retract of a paratopological group. We also notice, using some deep results of V.V. Uspenskij, that the space ω 1 of countable ordinals is not a retract of any topological group (see Fact 3 ). [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
27. The Baire theorem, an analogue of the Banach fixed point theorem and attractors in T1 compact spaces.
- Author
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Morayne, Michał and Rałowski, Robert
- Subjects
- *
COMPACT spaces (Topology) , *BAIRE spaces - Abstract
We prove that if X is a T 1 second countable compact space, then X is a Baire space if and only if every nonempty open subset of X contains a closed subset with nonempty interior. We also prove an analogue of Banach's fixed point theorem for all T 1 compact spaces. Applying the analogue of Banach's fixed point theorem we prove the existence of unique attractors for so called contractive iterated function systems whose Hutchinson operators are closed in compact T 1 spaces. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
28. Domination by a Polish space of the complement of the diagonal of X implies that X is cosmic.
- Author
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Guerrero Sánchez, David and Tkachuk, Vladimir V.
- Subjects
- *
DOMINATING set , *POLISH spaces (Mathematics) , *COMPACT spaces (Topology) , *CONTINUUM hypothesis , *TOPOLOGICAL spaces - Abstract
We will prove that a Tychonoff space X is analytic if and only if ( X × X ) \ Δ is dominated by a Polish space; here Δ = { ( x , x ) : x ∈ X } is the diagonal of X . This solves two published open questions. We will also establish under CH, that a Tychonoff space X has a countable network whenever ( X × X ) \ Δ is dominated by a second countable space. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
29. Function spaces jointly metrizable on compacta.
- Author
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Tkachuk, Vladimir V.
- Subjects
- *
FUNCTION spaces , *COMPACT spaces (Topology) , *SUBSET selection , *TOPOLOGICAL spaces , *MATHEMATICAL analysis - Abstract
If C p ( X ) is jointly metrizable on compacta, then p ( X ) ≤ ω but ω 1 need not be a caliber of X . If X is either submetrizable or a P -space, then C p ( C p ( X ) ) is jointly metrizable on compacta and, in particular, all compact subsets of C p ( C p ( X ) ) are metrizable. We show that for any dyadic compact X , the space C p ( X ) is jointly metrizable on compacta. Therefore, the JCM property of C p ( X ) for a compact space X does not imply that X is separable. If X is a compact space of countable tightness and C p ( X ) is jointly metrizable on compacta, then it is independent of ZFC whether X must be separable. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
30. CHAINS OF FUNCTIONS IN $C(K)$-SPACES.
- Author
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KANIA, TOMASZ and SMITH, RICHARD J.
- Subjects
- *
COMPACT spaces (Topology) , *COMPACT operators , *HAUSDORFF spaces , *LINEAR operators , *OPERATOR theory - Abstract
The Bishop property (♗), introduced recently by K. P. Hart, T. Kochanek and the first-named author, was motivated by Pełczyński’s classical work on weakly compact operators on $C(K)$-spaces. This property asserts that certain chains of functions in said spaces, with respect to a particular partial ordering, must be countable. There are two versions of (♗): one applies to linear operators on $C(K)$-spaces and the other to the compact Hausdorff spaces themselves. We answer two questions that arose after (♗) was first introduced. We show that if $\mathscr{D}$ is a class of compact spaces that is preserved when taking closed subspaces and Hausdorff quotients, and which contains no nonmetrizable linearly ordered space, then every member of $\mathscr{D}$ has (♗). Examples of such classes include all $K$ for which $C(K)$ is Lindelöf in the topology of pointwise convergence (for instance, all Corson compact spaces) and the class of Gruenhage compact spaces. We also show that the set of operators on a $C(K)$-space satisfying (♗) does not form a right ideal in $\mathscr{B}(C(K))$. Some results regarding local connectedness are also presented. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
31. DOMINATION CONDITIONS UNDER WHICH A COMPACT SPACE IS METRISABLE.
- Author
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DOW, ALAN and GUERRERO SÁNCHEZ, DAVID
- Subjects
- *
DOMINATING set , *SPACES of measures , *METRIC spaces , *METRIC geometry , *COMPACT spaces (Topology) - Abstract
In this note we partially answer a question of Cascales, Orihuela and Tkachuk [‘Domination by second countable spaces and Lindelöf ${\rm\Sigma}$-property’, Topology Appl.158(2) (2011), 204–214] by proving that under $CH$ a compact space $X$ is metrisable provided $X^{2}\setminus {\rm\Delta}$ can be covered by a family of compact sets $\{K_{f}:f\in {\it\omega}^{{\it\omega}}\}$ such that $K_{f}\subset K_{h}$ whenever $f\leq h$ coordinatewise. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
32. Maximal countably compact spaces and embeddings in MP-spaces.
- Author
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Tkachuk, V. and Wilson, R.
- Subjects
- *
EMBEDDINGS (Mathematics) , *COMPACT spaces (Topology) , *CUBES , *DISCONTINUOUS functions , *METAPHYSICS - Abstract
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] 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( X) has the Mazur property, then it is a Fréchet-Urysohn space. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
33. On the Bishop–Phelps–Bollobás property for numerical radius in spaces.
- Author
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Avilés, A., Guirao, A.J., and Rodríguez, J.
- Subjects
- *
COMPACT spaces (Topology) , *NUMERICAL analysis , *RADIUS (Geometry) , *BANACH spaces , *MATHEMATICAL analysis - Abstract
We study the Bishop–Phelps–Bollobás property for numerical radius within the framework of spaces. We present several sufficient conditions on a compact space K ensuring that has the Bishop–Phelps–Bollobás property for numerical radius. In particular, we show that has such property whenever K is metrizable. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
34. The cardinality of compact spaces satisfying the countable chain condition.
- Author
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Toshimichi Usuba
- Subjects
- *
COMPACT spaces (Topology) , *TOPOLOGICAL spaces , *HAUSDORFF spaces , *MATHEMATICAL analysis , *CARDINAL numbers , *TOPOLOGY - Abstract
We prove that for a compact Hausdorff space X, if λc (X) < w (X) for every infinite cardinal λ < w (X) and λc (X) < cf (w (X)) for every infinite cardinal λ < cf (w (X)), then Tikhonov cube [0,1] w (X) is a continuous image of X, in particular the cardinality of X is just 2 w (X). As an application of this result, we consider elementary submodel spaces and improve Tall's result in [17]. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
35. COUNTABLY Z-COMPACT SPACES.
- Author
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AL-ANI, A. T.
- Subjects
- *
COMPACT spaces (Topology) , *CONTINUOUS functions , *SET theory , *REALCOMPACT spaces , *TOPOLOGICAL spaces , *MATHEMATICAL analysis - Abstract
In this work we study countably z-compact spaces and z-Lindelof spaces. Several new properties of them are given. It is proved that every countably z-compact space is pseuodocompact (a space on which every real valued continuous function is bounded). Spaces which are countably z-compact but not countably compact are given. It is proved that a space is countably z-compact iff every countable z-closed set is compact. Characterizations of countably z-compact and z-Lindelof spaces by multifunctions are given. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
36. Both compact and sequentially compact sets in abelian topological group
- Author
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Li, Ronglu, Guo, Hao, and Swartz, C.
- Subjects
- *
COMPACT spaces (Topology) , *ABELIAN groups , *TOPOLOGICAL groups , *HAUSDORFF measures , *CONVEX functions , *STOCHASTIC convergence - Abstract
Abstract: We show that every abelian topological group contains many interesting sets which are both compact and sequentially compact. Then we can deduce some useful facts, e.g., [(1)] if G is a Hausdorff abelian topological group and is countably additive, then the range is compact metrizable; [(2)] if X is a Hausdorff locally convex space and , then is relatively compact in if and only if F is relatively compact in X, and if and only if F is relatively compact in where is the Dierolf topology which is the strongest -polar topology having the same subseries convergent series as the weak topology. [Copyright &y& Elsevier]
- Published
- 2011
- Full Text
- View/download PDF
37. Domination by second countable spaces and Lindelöf Σ-property
- Author
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Cascales, B., Orihuela, J., and Tkachuk, V.V.
- Subjects
- *
COMPACT spaces (Topology) , *SET theory , *POLISH spaces (Mathematics) , *METRIC spaces , *FUNCTION spaces , *TOPOLOGICAL spaces , *FUNCTIONAL analysis - Abstract
Abstract: Given a space M, a family of sets of a space X is ordered by M if { is a compact subset of M} and implies . We study the class of spaces which have compact covers ordered by a second countable space. We prove that a space belongs to if and only if it is a Lindelöf Σ-space. Under , if X is compact and has a compact cover ordered by a Polish space then X is metrizable; here is the diagonal of the space X. Besides, if X is a compact space of countable tightness and belongs to then X is metrizable in ZFC. We also consider the class of spaces X which have a compact cover ordered by a second countable space with the additional property that, for every compact set there exists with . It is a ZFC result that if X is a compact space and belongs to then X is metrizable. We also establish that, under CH, if X is compact and belongs to then X is countable. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
38. A glance at spaces with closure-preserving local bases
- Author
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Dow, A., Ramírez Martínez, R., and Tkachuk, V.V.
- Subjects
- *
COMPACT spaces (Topology) , *VECTOR topology , *SCATTERING (Mathematics) , *MATHEMATICAL continuum , *BASES (Linear topological spaces) , *MATHEMATICAL analysis - Abstract
Abstract: Call a space X (weakly) Japanese at a point if X has a closure-preserving local base (or quasi-base respectively) at the point x. The space X is (weakly) Japanese if it is (weakly) Japanese at every . We prove, in particular, that any generalized ordered space is Japanese and that the property of being (weakly) Japanese is preserved by σ-products; besides, a dyadic compact space is weakly Japanese if and only if it is metrizable. It turns out that every scattered Corson compact space is Japanese while there exist even Eberlein compact spaces which are not weakly Japanese. We show that a continuous image of a compact first countable space can fail to be weakly Japanese so the (weak) Japanese property is not preserved by perfect maps. Another interesting property of Japanese spaces is their tightness-monolithity, i.e., in every weakly Japanese space X we have for any set . [Copyright &y& Elsevier]
- Published
- 2010
- Full Text
- View/download PDF
39. On some questions about KC and related spaces
- Author
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Baldovino, Chiara and Costantini, Camillo
- Subjects
- *
COMPACT spaces (Topology) , *HAUSDORFF compactifications , *TOPOLOGICAL spaces , *ANALYTIC sets , *TOPOLOGY , *MATHEMATICS - Abstract
Abstract: Answering questions raised by O.T. Alas and R.G. Wilson, or by these two authors together with M.G. Tkachenko and V.V. Tkachuk, we show that every minimal SC space must be sequentially compact, and we produce the following examples: [–] a KC space which cannot be embedded in any compact KC space; [–] a countable KC space which does not admit any coarser compact KC topology; [–] a minimal Hausdorff space which is not a k-space. We also give an example of a compact KC space such that every nonempty open subset of it is dense, even if, as pointed out to us by the referee, a completely different construction carried out by E.K. van Douwen in 1993 leads to a space with the same properties. [Copyright &y& Elsevier]
- Published
- 2009
- Full Text
- View/download PDF
40. A net and open-filter process of compactification and the Stone–Čech, Wallman compactifications
- Author
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Wu, Hueytzen J. and Wu, Wan-Hong
- Subjects
- *
WALLMAN compactifications , *COMPACTIFICATION (Mathematics) , *NETS (Mathematics) , *COMPACT spaces (Topology) - Abstract
Abstract: By a characterization of compact spaces in Section 1, a process of obtaining a compactification of an arbitrary topological space X is described in Section 2 by a combined approach of nets and open filters. The Wallman compactification can be embedded in if X is Hausdorff and by a little modification, the compactification of X is the Stone–Čech compactification of X if X is Tychonoff. [Copyright &y& Elsevier]
- Published
- 2006
- Full Text
- View/download PDF
41. Comparability, Stability, and Completions of Ideals.
- Author
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Lu, Dancheng, Li, Qisheng, and Tong, Wenting
- Subjects
- *
IDEALS (Algebra) , *ASSOCIATIVE rings , *RING theory , *MATHEMATICAL functions , *ALGEBRA , *COMPACT spaces (Topology) - Abstract
In this paper, we study the properties of 1-comparability and stable range one condition on ideals of regular rings, and we use these results to investigate normalized pseudo-rank functions on ideals and N*-complete ideals. These will generalize the corresponding results of Goodearl. Finally, we give some sufficient conditions under which P(I) is compact. [ABSTRACT FROM AUTHOR]
- Published
- 2004
- Full Text
- View/download PDF
42. Problems on Universals
- Author
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Collins, P.J.
- Subjects
- *
TOPOLOGY , *COMPACT spaces (Topology) , *TOPOLOGICAL spaces , *SET theory - Abstract
Some problems arising out of recent work of P.M. Gartside and J.T.H. Lo are here surveyed. [Copyright &y& Elsevier]
- Published
- 2004
- Full Text
- View/download PDF
43. On the Weight of Nowhere Dense Subsets in Compact Spaces.
- Author
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Ivanov, A. V.
- Subjects
- *
COMPACT spaces (Topology) , *INVARIANTS (Mathematics) , *SET theory , *TOPOLOGICAL spaces , *MATHEMATICAL inequalities - Abstract
We study a new cardinal-valued invariant (calling it the nd-weight of X) of a topological space which is defined as the least upper bound of the weights of nowhere dense subsets of X. The main result is the proof of the inequality hl(X)≤ndw(X) for compact sets without isolated points ((hl is the hereditary Lindelof number). This inequality implies that a compact space without isolated points of countable nd-weight is completely normal. Assuming the continuum hypothesis, we construct an example of a nonmetrizable compact space of countable nd-weight without isolated points. [ABSTRACT FROM AUTHOR]
- Published
- 2003
- Full Text
- View/download PDF
44. Remarks on the set of -points in Eberlein and Corson compact spaces
- Author
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Krawczyk, Adam, Marciszewski, Witold, and Michalewski, Henryk
- Subjects
- *
COMPACT spaces (Topology) , *TOPOLOGICAL spaces , *MATHEMATICAL programming , *ALGEBRAIC topology , *SET theory - Abstract
Abstract: For a compact space K we consider the set of all -points in K, i.e., points with countable base of neighborhoods in K. We show that, for every scattered Eberlein compact space K, the set is a -set in K. We also give an example of a scattered Eberlein compactum with non-metrizable set . Moreover, we give an example of a Corson compact space K such that does not contain any dense -subset of K. This answers three questions of Tkachuk. [Copyright &y& Elsevier]
- Published
- 2009
- Full Text
- View/download PDF
45. A strengthening of the Čech–Pospišil theorem
- Author
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Juhász, István and Szentmiklóssy, Zoltán
- Subjects
- *
TOPOLOGICAL graph theory , *COMPACT spaces (Topology) , *DISCRETE groups , *COVERING spaces (Topology) , *SET theory , *ALGEBRAIC topology - Abstract
Abstract: We prove the following result: If in a compact space X there is a λ-branching family of closed sets then X cannot be covered by fewer than λ many discrete subspaces. (A family of sets is λ-branching iff but one can form λ many pairwise disjoint intersections of subfamilies of .) The proof is based on a recent, still unpublished, lemma of G. Gruenhage. As a consequence, we obtain the following strengthening of the well-known Čech–Pospišil theorem: If X a is compact space such that all points have character then X cannot be covered by fewer than many discrete subspaces. [Copyright &y& Elsevier]
- Published
- 2008
- Full Text
- View/download PDF
46. Cellularity of infinite Hausdorff spaces in ZF.
- Author
-
Keremedis, Kyriakos and Tachtsis, Eleftherios
- Subjects
- *
HAUSDORFF spaces , *SET theory , *OPEN-ended questions , *BOOLEAN algebra , *COMPACT spaces (Topology) - Abstract
In set theory without the Axiom of Choice (AC), we investigate the set-theoretic strength (in terms of weak choice principles) of the following statements: "every infinite Hausdorff space has a denumerable cellular family", "every infinite Hausdorff space has a denumerable discrete subset", "every denumerable compact Hausdorff space has an infinite cellular family", and "every denumerable compact Hausdorff space has an infinite discrete subset", and answer open questions by E. Tachtsis "Infinite Hausdorff spaces may lack cellular families or discrete subsets of cardinality ℵ 0 " and an open question by A. Miller "Some interesting problems", which also appears in the former paper. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
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