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1
Kovár, Martin Maria. "The Classes of Mutual Compactificability." International Journal of Mathematics and Mathematical Sciences 2007 (2007): 1–11. http://dx.doi.org/10.1155/2007/16135.
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Two disjoint topological spacesX,Yare mutually compactificable if there exists a compact topology onK=X∪Ywhich coincides onX,Ywith their original topologies such that the pointsx∈X,y∈Yhave disjoint neighborhoods inK. The main problem under consideration is the following: which spacesX,Yare so compatible such that they together can form the compact spaceK? In this paper we define and study the classes of spaces with the similar behavior with respect to the mutual compactificability. Two spacesX1,X2belong to the same class if they can substitute each other in the above construction with any spaceY. In this way we transform the main problem to the study of relations between the compactificability classes. Some conspicuous classes of topological spaces are discovered as the classes of mutual compactificability. The studied classes form a certain “scale of noncompactness” for topological spaces. Every class of mutual compactificability contains aT1representative, but there are classes with no Hausdorff representatives.
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Ozturk, Taha Yasin, and Sadi Bayramov. "Soft Mappings Space." Scientific World Journal 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/307292.
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Various soft topologies are being introduced on a given function space soft topological spaces. In this paper, soft compact-open topology is defined in functional spaces of soft topological spaces. Further, these functional spaces are studied and interrelations between various functional spaces with soft compact-open topology are established.
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BHATTACHARJEE, PAPIYA. "TWO SPACES OF MINIMAL PRIMES." Journal of Algebra and Its Applications 11, no. 01 (February 2012): 1250014. http://dx.doi.org/10.1142/s0219498811005373.
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This paper studies algebraic frames L and the set Min (L) of minimal prime elements of L. We will endow the set Min (L) with two well-known topologies, known as the Hull-kernel (or Zariski) topology and the inverse topology, and discuss several properties of these two spaces. It will be shown that Min (L) endowed with the Hull-kernel topology is a zero-dimensional, Hausdorff space; whereas, Min (L) endowed with the inverse topology is a T1, compact space. The main goal will be to find conditions on L for the spaces Min (L) and Min (L)-1 to have various topological properties; for example, compact, locally compact, Hausdorff, zero-dimensional, and extremally disconnected. We will also discuss when the two topological spaces are Boolean and Stone spaces.
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Aurichi, Leandro F., and Rodrigo R. Dias. "Topological Games and Alster Spaces." Canadian Mathematical Bulletin 57, no. 4 (December 1, 2014): 683–96. http://dx.doi.org/10.4153/cmb-2013-048-5.
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AbstractIn this paper we study connections between topological games such as Rothberger, Menger, and compact-open games, and we relate these games to properties involving covers byGδsubsets. The results include the following: (1) If TWO has a winning strategy in theMenger game on a regular spaceX, thenXis an Alster space. (2) If TWO has a winning strategy in the Rothberger game on a topological spaceX, then theGδ-topology onXis Lindelöf. (3) The Menger game and the compact-open game are (consistently) not dual.
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Georgiou, D. N., and B. K. Papadopoulos. "On nearly compact topological and fuzzy topological spaces." Topology and its Applications 123, no. 1 (August 2002): 73–85. http://dx.doi.org/10.1016/s0166-8641(01)00171-7.
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Bardyla, Serhii, and Alex Ravsky. "Closed subsets of compact-like topological spaces." Applied General Topology 21, no. 2 (October 1, 2020): 201. http://dx.doi.org/10.4995/agt.2020.12258.
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<p>We investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces (semigroups, resp.). We show that each Hausdorff topological space is a closed subspace of some Hausdorff ω-bounded pracompact topological space and describe open dense subspaces of<br />countably pracompact topological spaces. We construct a pseudocompact topological semigroup which contains the bicyclic monoid as a closed subsemigroup. This example provides an affirmative answer to a question posed by Banakh, Dimitrova, and Gutik in [4]. Also, we show that the semigroup of ω×ω-matrix units cannot be embedded into a Hausdorff topological semigroup whose space is weakly H-closed.</p>
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Gabriyelyan, Saak S., and Sidney A. Morris. "Embedding into free topological vector spaces on compact metrizable spaces." Topology and its Applications 233 (January 2018): 33–43. http://dx.doi.org/10.1016/j.topol.2017.09.008.
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Coquand, Thierry. "Minimal invariant spaces in formal topology." Journal of Symbolic Logic 62, no. 3 (September 1997): 689–98. http://dx.doi.org/10.2307/2275567.
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A standard result in topological dynamics is the existence of minimal subsystem. It is a direct consequence of Zorn's lemma: given a compact topological space X with a map f: X→X, the set of compact non empty subspaces K of X such that f(K) ⊆ K ordered by inclusion is inductive, and hence has minimal elements. It is natural to ask for a point-free (or formal) formulation of this statement. In a previous work [3], we gave such a formulation for a quite special instance of this statement, which is used in proving a purely combinatorial theorem (van de Waerden's theorem on arithmetical progression).In this paper, we extend our analysis to the case where X is a boolean space, that is compact totally disconnected. In such a case, we give a point-free formulation of the existence of a minimal subspace for any continuous map f: X→X. We show that such minimal subspaces can be described as points of a suitable formal topology, and the “existence” of such points become the problem of the consistency of the theory describing a generic point of this space. We show the consistency of this theory by building effectively and algebraically a topological model. As an application, we get a new, purely algebraic proof, of the minimal property of [3]. We show then in detail how this property can be used to give a proof of (a special case of) van der Waerden's theorem on arithmetical progression, that is “similar in structure” to the topological proof [6, 8], but which uses a simple algebraic remark (Proposition 1) instead of Zorn's lemma. A last section tries to place this work in a wider context, as a reformulation of Hilbert's method of introduction/elimination of ideal elements.
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Al-shami, T. M. "Sum of the spaces on ordered setting." Moroccan Journal of Pure and Applied Analysis 6, no. 2 (December 1, 2020): 255–65. http://dx.doi.org/10.2478/mjpaa-2020-0020.
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AbstractOne of the divergences between topology and ordered topology is that some topological concepts such as separation axioms and continuous maps are defined using open neighborhoods or neighborhoods without any difference, however, they are distinct on the ordered topology according to the neighborhoods: Are they open neighborhoods or not? In this paper, we present the concept of sum of the ordered spaces using pairwise disjoint topological ordered spaces and study main properties. Then, we introduce the properties of ordered additive, finitely ordered additive and countably ordered additive which associate topological ordered spaces with their sum. We prove that the properties of being Ti-ordered and strong Ti-ordered spaces are ordered additive, however, the properties of monotonically compact and ordered compact spaces are finitely ordered additive. Also, we define a mapping between two sums of the ordered spaces using mappings between the ordered spaces and deduce some results related to some types of continuity and homeomorphism. We complete this work by determining the conditions under which a topological ordered space is sum of the ordered spaces.
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Latif, Raja Mohammad. "Infra -α- Compact and Infra -α- Connected Spaces." International Journal of Pure Mathematics 8 (August 11, 2021): 41–57. http://dx.doi.org/10.46300/91019.2021.8.6.
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In 2016 Hakeem A. Othman and Md. Hanif Page introduced a new notion of set in general topology called an infra -α- open set and investigated its fundamental properties and studied the relationship between infra -α- open set and other topological sets. The objective of this paper is to introduce the new concepts called infra -α- compact space, countably infra -α- compact space, infra -α- Lindelof space, almost infra -α- compact space, mildly infra -α- compact space and infra -α- connected space in general topology and investigate several properties and characterizations of these new concepts in topological spaces.
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Kovár, Martin Maria. "Mutually Compactificable Topological Spaces." International Journal of Mathematics and Mathematical Sciences 2007 (2007): 1–10. http://dx.doi.org/10.1155/2007/70671.
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Two disjoint topological spacesX,Yare(T2-)mutually compactificable if there exists a compact(T2-)topology onK=X∪Ywhich coincides onX,Ywith their original topologies such that the pointsx∈X,y∈Yhave open disjoint neighborhoods inK. This paper, the first one from a series, contains some initial investigations of the notion. Some key properties are the following: a topological space is mutually compactificable with some space if and only if it isθ-regular. A regular space on which every real-valued continuous function is constant is mutually compactificable with noS2-space. On the other hand, there exists a regular non-T3.5space which is mutually compactificable with the infinite countable discrete space.
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MAGHSOUDI, SAEID, and RASOUL NASR-ISFAHANI. "STRICT TOPOLOGY AS A MIXED TOPOLOGY ON LEBESGUE SPACES." Bulletin of the Australian Mathematical Society 84, no. 3 (September 6, 2011): 504–15. http://dx.doi.org/10.1017/s0004972711002589.
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AbstractLetXbe a locally compact space, and 𝔏∞0(X,ι) be the space of all essentially boundedι-measurable functionsfonXvanishing at infinity. We introduce and study a locally convex topologyβ1(X,ι) on the Lebesgue space 𝔏1(X,ι) such that the strong dual of (𝔏1(X,ι),β1(X,ι)) can be identified with$({\frak L}_0^\infty (X,\iota ),\|\cdot \|_\infty )$. Next, by showing thatβ1(X,ι) can be considered as a natural mixed topology, we deduce some of its basic properties. Finally, as an application, we prove thatL1(G) , the group algebra of a locally compact Hausdorff topological groupG, equipped with the convolution multiplication is a complete semitopological algebra under theβ1(G) topology.
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Lipparini, Paolo. "Compact factors in finally compact products of topological spaces." Topology and its Applications 153, no. 9 (March 2006): 1365–82. http://dx.doi.org/10.1016/j.topol.2005.04.002.
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Elmali, Ceren Sultan, and Tamer Ugŭr. "Fan-Gottesman Compactification and Scattered Spaces." Applied Mathematics and Nonlinear Sciences 5, no. 1 (April 10, 2020): 475–78. http://dx.doi.org/10.2478/amns.2020.1.00045.
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AbstractCompactification is the process or result of making a topological space into a compact space. An embedding of a topological space X as a dense subset of a compact space is called a compactification of X. There are a lot of compactification methods but we study with Fan- Gottesman compactification. A topological space X is said to be scattered if every nonempty subset S of X contains at least one point which is isolated in S. Compact scattered spaces are important for analysis and topology. In this paper, we investigate the relation between the Fan-Gottesman compactification of T3 space and scattered spaces. We show under which conditions the Fan-Gottesman compactification X* is a scattered.
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Cao, Jiling, and David Gauld. "Volterra spaces revisited." Journal of the Australian Mathematical Society 79, no. 1 (August 2005): 61–76. http://dx.doi.org/10.1017/s1446788700009332.
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AbstractIn this paper, we investigate Volterra spaces and relevant topological properties. New characterizations of weakly Volterra spaces are provided. An analogy of the Banach category theorem in terms of Volterra properties is obtained. It is shown that every weakly Volterra homogeneous space is Volterra, and there are metrizable Baire spaces whose hyperspaces of nonempty compact subsets endowed with the Vietoris topology are not weakly Volterra.
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Flum, Jörg, and Juan Carlos Martinez. "On topological spaces equivalent to ordinals." Journal of Symbolic Logic 53, no. 3 (September 1988): 785–95. http://dx.doi.org/10.2307/2274571.
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AbstractLet L be one of the topological languages Lt, (L∞ω)t and (Lκω)t. We characterize the topological spaces which are models of the L-theory of the class of ordinals equipped with the order topology. The results show that the role played in classical model theory by the property of being well-ordered is taken over in the topological context by the property of being locally compact and scattered.
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Smyth, M. B., and J. Webster. "Finite approximation of stably compact spaces." Applied General Topology 3, no. 2 (October 1, 2002): 197. http://dx.doi.org/10.4995/agt.2002.2063.
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<p>Finite approximation of spaces by inverse sequences of graphs (in the category of so-called topological graphs) was introduced by Smyth, and developed further. The idea was subsequently taken up by Kopperman and Wilson, who developed their own purely topological approach using inverse spectra of finite T<sub>0</sub>-spaces in the category of stably compact spaces. Both approaches are, however, restricted to the approximation of (compact) Hausdorff spaces and therefore cannot accommodate, for example, the upper space and (multi-) function space constructions. We present a new method of finite approximation of stably compact spaces using finite stably compact graphs, which when the topology is discrete are simply finite directed graphs. As an extended example, illustrating the problems involved, we study (ordered spaces and) arcs.</p>
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Lin, Shou, Kedian Li, and Ying Ge. "On the metrizability of TVS-cone metric spaces." Publications de l'Institut Math?matique (Belgrade) 98, no. 112 (2015): 271–79. http://dx.doi.org/10.2298/pim1512271l.
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Metric spaces are cone metric spaces, and cone metric spaces are TVS-cone metric spaces. We prove that TVS-cone metric spaces are paracompact. A metrization theorem of TVS-cone metric spaces is obtained by a purely topological tools. We obtain that a homeomorphism f of a compact space is expansive if and only if f is TVS-cone expansive. In the end, for a TVS-cone metric topology, a concrete metric generating the topology is constructed.
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Castellano, Ilaria, and Anna Giordano Bruno. "Topological entropy for locally linearly compact vector spaces." Topology and its Applications 252 (February 2019): 112–44. http://dx.doi.org/10.1016/j.topol.2018.11.009.
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Brattka, Vasco. "Effective representations of the space of linear bounded operators." Applied General Topology 4, no. 1 (April 1, 2003): 115. http://dx.doi.org/10.4995/agt.2003.2014.
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<p>Representations of topological spaces by infinite sequences of symbols are used in computable analysis to describe computations in topological spaces with the help of Turing machines. From the computer science point of view such representations can be considered as data structures of topological spaces. Formally, a representation of a topological space is a surjective mapping from Cantor space onto the corresponding space. Typically, one is interested in admissible, i.e. topologically well-behaved representations which are continuous and characterized by a certain maximality condition. We discuss a number of representations of the space of linear bounded operators on a Banach space. Since the operator norm topology of the operator space is nonseparable in typical cases, the operator space cannot be represented admissibly with respect to this topology. However, other topologies, like the compact open topology and the Fell topology (on the operator graph) give rise to a number of promising representations of operator spaces which can partially replace the operator norm topology. These representations reflect the information which is included in certain data structures for operators, such as programs or enumerations of graphs. We investigate the sublattice of these representations with respect to continuous and computable reducibility. Certain additional conditions, such as finite dimensionality, let some classes of representations collapse, and thus, change the corresponding graph. Altogether, a precise picture of possible data structures for operator spaces and their mutual relation can be drawn.</p>
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Hoffmann, Rudolf-E. "The Injective Hull and the -Compactification of a Continuous Poset." Canadian Journal of Mathematics 37, no. 5 (October 1, 1985): 810–53. http://dx.doi.org/10.4153/cjm-1985-045-3.
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In [57] (2.12), D. S. Scott showed that the continuous lattices, invented by him in his study of a mathematical theory of computation [56], are precisely (when they are made into topological spaces via the Scott topology) the injective T0-spaces, i.e., the injective objects in the category T0 of T0-spaces and continuous maps with regard to the class of all embeddings. Moreover, the sort of morphisms between continuous lattices Scott considered are precisely the continuous maps with regard to the respective Scott topologies. These are fairly non-Hausdorff topologies. (Indeed, the Scott topology induces the partial order of the lattice L via x ≦ y if and only if x ∊ cl{j}, the “specialization order” of the topology; hence L is Hausdorff in the Scott topology if and only if L has at most one element.) In topological algebra, compact Lawson semilattices (= compact Hausdorff topological ∧-semilattices such that the ∧-preserving continuous maps into the unit interval, with its ordinary topology and the min-semilattice structure, separate the points) with a unit element 1 have attracted considerable interest.
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El-Shafei, M. E., M. Abo-Elhamayel, and T. M. Al-Shami. "Partial soft separation axioms and soft compact spaces." Filomat 32, no. 13 (2018): 4755–71. http://dx.doi.org/10.2298/fil1813755e.
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The main aim of the present paper is to define new soft separation axioms which lead us, first, to generalize existing comparable properties via general topology, second, to eliminate restrictions on the shape of soft open sets on soft regular spaces which given in [22], and third, to obtain a relationship between soft Hausdorff and new soft regular spaces similar to those exists via general topology. To this end, we define partial belong and total non belong relations, and investigate many properties related to these two relations. We then introduce new soft separation axioms, namely p-soft Ti-spaces (i = 0,1,2,3,4), depending on a total non belong relation, and study their features in detail. With the help of examples, we illustrate the relationships among these soft separation axioms and point out that p-soft Ti-spaces are stronger than soft Ti-spaces, for i = 0,1,4. Also, we define a p-soft regular space, which is weaker than a soft regular space and verify that a p-soft regular condition is sufficient for the equivalent among p-soft Ti-spaces, for i = 0,1,2. Furthermore, we prove the equivalent among finite p-soft Ti-spaces, for i = 1,2,3 and derive that a finite product of p-soft Ti-spaces is p-soft Ti, for i = 0,1,2,3,4. In the last section, we show the relationships which associate some p-soft Ti-spaces with soft compactness, and in particular, we conclude under what conditions a soft subset of a p-soft T2-space is soft compact and prove that every soft compact p-soft T2-space is soft T3-space. Finally, we illuminate that some findings obtained in general topology are not true concerning soft topological spaces which among of them a finite soft topological space need not be soft compact.
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Mitra, Biswajit, and Debojyoti Chowdhury. "Ideal spaces." Applied General Topology 22, no. 1 (April 1, 2021): 79. http://dx.doi.org/10.4995/agt.2021.13608.
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<p>Let C<sub>∞ </sub>(X) denote the family of real-valued continuous functions which vanish at infinity in the sense that {x ∈ X : |f(x)| ≥ 1/n} is compact in X for all n ∈ N. It is not in general true that C<span style="vertical-align: sub;">∞ </span>(X) is an ideal of C(X). We define those spaces X to be ideal space where C<span style="vertical-align: sub;">∞ </span>(X) is an ideal of C(X). We have proved that nearly pseudocompact spaces are ideal spaces. For the converse, we introduced a property called “RCC” property and showed that an ideal space X is nearly pseudocompact if and only if X satisfies ”RCC” property. We further discussed some topological properties of ideal spaces.</p>
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Al-shami, Tareq M. "Infra Soft Compact Spaces and Application to Fixed Point Theorem." Journal of Function Spaces 2021 (July 15, 2021): 1–9. http://dx.doi.org/10.1155/2021/3417096.
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Infra soft topology is one of the recent generalizations of soft topology which is closed under finite intersection. Herein, we contribute to this structure by presenting two kinds of soft covering properties, namely, infra soft compact and infra soft Lindelöf spaces. We describe them using a family of infra soft closed sets and display their main properties. With the assistance of examples, we mention some classical topological properties that are invalid in the frame of infra soft topology and determine under which condition they are valid. We focus on studying the “transmission” of these concepts between infra soft topology and classical infra topology which helps us to discover the behaviours of these concepts in infra soft topology using their counterparts in classical infra topology and vice versa. Among the obtained results, these concepts are closed under infra soft homeomorphisms and finite product of soft spaces. Finally, we introduce the concept of fixed soft points and reveal main characterizations, especially those induced from infra soft compact spaces.
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Smrekar, Jaka. "Homotopy Characterization of ANR Function Spaces." Journal of Function Spaces and Applications 2013 (2013): 1–5. http://dx.doi.org/10.1155/2013/925742.
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LetYbe an absolute neighbourhood retract (ANR) for the class of metric spaces and letXbe a topological space. LetYXdenote the space of continuous maps fromXtoYequipped with the compact open topology. We show that ifXis a compactly generated Tychonoff space andYis not discrete, thenYXis an ANR for metric spaces if and only ifXis hemicompact andYXhas the homotopy type of a CW complex.
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WANG, HANFENG, and WEI HE. "A NOTE ON -SPACES." Bulletin of the Australian Mathematical Society 90, no. 1 (May 12, 2014): 144–48. http://dx.doi.org/10.1017/s0004972714000112.
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AbstractIn this paper, it is shown that every compact Hausdorff $K$-space has countable tightness. This result gives a positive answer to a problem posed by Malykhin and Tironi [‘Weakly Fréchet–Urysohn and Pytkeev spaces’, Topology Appl.104 (2000), 181–190]. We show that a semitopological group $G$ that is a $K$-space is first countable if and only if $G$ is of point-countable type. It is proved that if a topological group $G$ is a $K$-space and has a locally paracompact remainder in some Hausdorff compactification, then $G$ is metrisable.
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Čerin, Zvonko. "Proximate topology and shape theory." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 125, no. 3 (1995): 595–615. http://dx.doi.org/10.1017/s0308210500032704.
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Most of the development of shape theory was in the so-called outer shape theory, where the shape of spaces is described with the help of some outside objects.This paper belongs to the so-called inner shape theory, in which the shape of spaces is described intrinsically without the use of any outside gadgets. We give a description of shape theory that does not need absolute neighbourhood retracts. We prove that the category ℋN whose objects are topological spaces and whose morphisms are proximate homotopy classes of proximate nets is naturally equivalent to the shape category h. The description of the category ℋN for compact metric spaces was given earlier by José M. R. Sanjurjo. We also give three applications of this new approach to shape theory.
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Mehmood, Arif, Saleem Abdullah, Mohammed M. Al-Shomrani, Muhammad Imran Khan, and Orawit Thinnukool. "Some Results in Neutrosophic Soft Topology Concerning Neutrosophic Soft ∗ b Open Sets." Journal of Function Spaces 2021 (May 24, 2021): 1–15. http://dx.doi.org/10.1155/2021/5544319.
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In this article, new generalised neutrosophic soft open known as neutrosophic soft ∗ b open set is introduced in neutrosophic soft topological spaces. Neutrosophic soft ∗ b open set is generated with the help of neutrosophic soft semiopen and neutrosophic soft preopen sets. Then, with the application of this new definition, some soft neutrosophical separation axioms, countability theorems, and countable space can be Hausdorff space under the subjection of neutrosophic soft sequence which is convergent, the cardinality of neutrosophic soft countable space, engagement of neutrosophic soft countable and uncountable spaces, neutrosophic soft topological features of the various spaces, soft neutrosophical continuity, the product of different soft neutrosophical spaces, and neutrosophic soft countably compact that has the characteristics of Bolzano Weierstrass Property (BVP) are studied. In addition to this, BVP shifting from one space to another through neutrosophic soft continuous functions, neutrosophic soft sequence convergence, and its marriage with neutrosophic soft compact space, sequentially compactness are addressed.
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Corob Cook, Ged. "Eilenberg–Mac Lane Spaces for Topological Groups." Axioms 8, no. 3 (July 27, 2019): 90. http://dx.doi.org/10.3390/axioms8030090.
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In this paper, we establish a topological version of the notion of an Eilenberg–Mac Lane space. If X is a pointed topological space, π 1 ( X ) has a natural topology coming from the compact-open topology on the space of maps S 1 → X . In general, the construction does not produce a topological group because it is possible to create examples where the group multiplication π 1 ( X ) × π 1 ( X ) → π 1 ( X ) is discontinuous. This discontinuity has been noticed by others, for example Fabel. However, if we work in the category of compactly generated, weakly Hausdorff spaces, we may retopologise both the space of maps S 1 → X and the product π 1 ( X ) × π 1 ( X ) with compactly generated topologies to see that π 1 ( X ) is a group object in this category. Such group objects are known as k-groups. Next we construct the Eilenberg–Mac Lane space K ( G , 1 ) for any totally path-disconnected k-group G. The main point of this paper is to show that, for such a G, π 1 ( K ( G , 1 ) ) is isomorphic to G in the category of k-groups. All totally disconnected locally compact groups are k-groups and so our results apply in particular to profinite groups, answering a question of Sauer’s. We also show that analogues of the Mayer–Vietoris sequence and Seifert–van Kampen theorem hold in this context. The theory requires a careful analysis using model structures and other homotopical structures on cartesian closed categories as we shall see that no theory can be comfortably developed in the classical world.
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Perez-Garcia, Maria Cristina. "k-spaces and duals of non-archimedean metrizable locally convex spaces." Forum Mathematicum 30, no. 5 (September 1, 2018): 1309–18. http://dx.doi.org/10.1515/forum-2016-0080.
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AbstractThe main purpose of this paper is to investigate the non-archimedean counterpart of the classical result stating that the dual of a real or complex metrizable locally convex space, equipped with the locally convex topology of uniform convergence on compact sets, belongs to the topological category formed by the k-spaces. We prove that this counterpart holds when the non-archimedean valued base field {\mathbb{K}} is locally compact, but fails for any non-locally compact {\mathbb{K}}. Here we deal with a topological subcategory, the one formed by the {k_{0}}-spaces, the adequate non-archimedean substitutes for k-spaces. As a product, we complete some of the achievements on the non-archimedean Banach–Dieudonné Theorem presented in [C. Perez-Garcia and W. H. Schikhof, The p-adic Banach–Dieudonné theorem and semi-compact inductive limits, p-adic Functional Analysis (Poznań 1998), Lecture Notes Pure Appl. Math. 207, Dekker, New York 1999, 295–307]. Also, we use our results to construct in a simple way natural examples of k-spaces (which are also {k_{0}}-spaces) whose products are not {k_{0}}-spaces. This in turn improves the, rather involved, example given in [C. Perez-Garcia and W. H. Schikhof, Locally Convex Spaces over non-Archimedean Valued Fields, Cambridge Stud. Adv. Math. 119, Cambridge University Press, Cambridge, 2010] of two {k_{0}}-spaces whose product is not a {k_{0}}-space. Our theory covers an important class of non-archimedean Fréchet spaces, the Köthe sequence spaces, which have a relevant influence on applications such as the definition of a non-archimedean Laplace and Fourier transform.
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Lyakhovets, Daniil, and Alexander Osipov. "Selection principles and games in bitopological function spaces." Filomat 33, no. 14 (2019): 4535–40. http://dx.doi.org/10.2298/fil1914535l.
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For a Tychonoff space X, we denote by (C(X), ?k ?p) the bitopological space of all real-valued continuous functions on X, where ?k is the compact-open topology and ?p is the topology of pointwise convergence. In the papers [6, 7, 13] variations of selective separability and tightness in (C(X),?k,?p) were investigated. In this paper we continue to study the selective properties and the corresponding topological games in the space (C(X),?k,?p).
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Holdon, Liviu-Constantin. "The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices." Open Mathematics 18, no. 1 (November 6, 2020): 1206–26. http://dx.doi.org/10.1515/math-2020-0061.
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Abstract In this paper, by using the ideal theory in residuated lattices, we construct the prime and maximal spectra (Zariski topology), proving that the prime and maximal spectra are compact topological spaces, and in the case of De Morgan residuated lattices they become compact {T}_{0} topological spaces. At the same time, we define and study the reticulation functor between De Morgan residuated lattices and bounded distributive lattices. Moreover, we study the I-topology (I comes from ideal) and the stable topology and we define the concept of pure ideal. We conclude that the I-topology is in fact the restriction of Zariski topology to the lattice of ideals, but we use it for simplicity. Finally, based on pure ideals, we define the normal De Morgan residuated lattice (L is normal iff every proper ideal of L is a pure ideal) and we offer some characterizations.
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Akbarbaglu, I., and S. Maghsoudi. "On the Generalized Weighted Lebesgue Spaces of Locally Compact Groups." Abstract and Applied Analysis 2011 (2011): 1–15. http://dx.doi.org/10.1155/2011/947908.
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Let be a locally compact group with a fixed left Haar measure and be a system of weights on . In this paper, we deal with locally convex space equipped with the locally convex topology generated by the family of norms . We study various algebraic and topological properties of the locally convex space . In particular, we characterize its dual space and show that it is a semireflexive space. Finally, we give some conditions under which with the convolution multiplication is a topological algebra and then characterize its closed ideals and its spectrum.
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ECHI, OTHMAN, and MOHAMED OUELD ABDALLAHI. "ON THE SPECTRALIFICATION OF A HEMISPECTRAL SPACE." Journal of Algebra and Its Applications 10, no. 04 (August 2011): 687–99. http://dx.doi.org/10.1142/s0219498811004847.
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An open subset U of a topological space X is called intersection compact open, or ICO, if for every compact open set Q of X, U ∩ Q is compact. A continuous map f of topological spaces will be called spectral if f-1 carries ICO sets to ICO sets. Call a topological space Xhemispectral, if the intersection of two ICO sets of X is an ICO. Let HSPEC be the category whose objects are hemispectral spaces and arrows spectral maps. Let SPEC be the full subcategory of HSPEC whose objects are spectral spaces. The main result of this paper proves that SPEC is a reflective subcategory of HSPEC. This gives a complete answer to Problem BST1 of "O. Echi, H. Marzougui and E. Salhi, Problems from the Bizerte–Sfax–Tunis seminar, in Open Problems in Topology II, ed. E. Pearl (Elsevier, 2007), pp. 669–674."
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Geske, Christian. "Algebraic intersection spaces." Journal of Topology and Analysis 12, no. 04 (January 9, 2019): 1157–94. http://dx.doi.org/10.1142/s1793525319500778.
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We define a variant of intersection space theory that applies to many compact complex and real analytic spaces [Formula: see text], including all complex projective varieties; this is a significant extension to a theory which has so far only been shown to apply to a particular subclass of spaces with smooth singular sets. We verify existence of these so-called algebraic intersection spaces and show that they are the (reduced) chain complexes of known topological intersection spaces in the case that both exist. We next analyze “local duality obstructions,” which we can choose to vanish, and verify that algebraic intersection spaces satisfy duality in the absence of these obstructions. We conclude by defining an untwisted algebraic intersection space pairing, whose signature is equal to the Novikov signature of the complement in [Formula: see text] of a tubular neighborhood of the singular set.
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Banakh, Taras. "On κ-bounded and M-compact reflections of topological spaces." Topology and its Applications 289 (February 2021): 107547. http://dx.doi.org/10.1016/j.topol.2020.107547.
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Moradi, Hamid Reza. "Bounded and Semi Bounded Inverse Theorems in Fuzzy Normed Spaces." International Journal of Fuzzy System Applications 4, no. 2 (April 2015): 47–55. http://dx.doi.org/10.4018/ijfsa.2015040104.
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In this paper, the author introduces the notion of the complete fuzzy norm on a linear space. And the author considers some relations between the fuzzy completeness and ordinary completeness on a linear space, moreover a new form of fuzzy compact spaces, namely b-compact spaces, b-closed space is introduced. Some characterization of their properties is obtained. Also some basic properties for linear operators between fuzzy normed spaces are further studied. The notions of fuzzy vector spaces and fuzzy topological vector spaces were introduced in Katsaras and Liu (1977). These ideas were modified by Katsaras (1981), and in (1984) Katsaras defined the fuzzy norm on a vector space. In (1991) Krishna and Sarma discussed the generation of a fuzzy vector topology from an ordinary vector topology on vector spaces. Also Krishna and Sarma (1992) observed the convergence of sequence of fuzzy points. Rhie et al. (1997) Introduced the notion of fuzzy a-Cauchy sequence of fuzzy points and fuzzy completeness.
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Park, Sehie. "Best approximation theorems for composites of upper semicontinuous maps." Bulletin of the Australian Mathematical Society 51, no. 2 (April 1995): 263–72. http://dx.doi.org/10.1017/s000497270001409x.
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Let (E, τ) be a Hausdorff topological vector space and (X, ω) a weakly compact convex subset of E with the relative weak topology ω. Recently, there have appeared best approximation and fixed point theorems for convex-valued upper semicontinuous maps F: (X, ω) → 2(E, τ) whenever (E, τ) is locally convex. In this paper, these results are extended to a very broad class of multifunctions containing composites of acyclic maps in a topological vector space having sufficiently many linear functionals. Moreover, we also obtain best approximation theorems for classes of multifunctions defined on approximatively compact convex subsets of locally convex Hausdorff topological vector spaces or closed convex subsets of Banach spaces with the Oshman property.
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Boyd, C. "Exponential Laws for the Nachbin Ported Topology." Canadian Mathematical Bulletin 43, no. 2 (June 1, 2000): 138–44. http://dx.doi.org/10.4153/cmb-2000-021-x.
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AbstractWe show that for U and V balanced open subsets of (Qno) Fréchet spaces E and F that we have the topological identityAnalogous results for the compact open topology have long been established. We also give an example to show that the (Qno) hypothesis on both E and F is necessary.
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VILLENA, A. R. "UNIQUENESS OF THE TOPOLOGY ON SPACES OF VECTOR-VALUED FUNCTIONS." Journal of the London Mathematical Society 64, no. 2 (October 2001): 445–56. http://dx.doi.org/10.1112/s0024610701002423.
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Let Ω be a topological space without isolated points, let E be a topological linear space which is continuously embedded into a product of countably boundedly generated topological linear spaces, and let X be a linear subspace of C(Ω, E). If a ∈ C(Ω) is not constant on any open subset of Ω and aX ⊂ X, then it is shown that there is at most one F-space topology on X that makes the multiplication by a continuous. Furthermore, if [Ufr ] is a subset of C(Ω) which separates strongly the points of Ω and [Ufr ]X ⊂ X, then it is proved that there is at most one F-space topology on X that makes the multiplication by a continuous for each a ∈ [Ufr ].These results are applied to the study of the uniqueness of the F-space topology and the continuity of translation invariant operators on the Banach space L1(G, E) for a noncompact locally compact group G and a Banach space E. Furthermore, the problems of the uniqueness of the F-algebra topology and the continuity of epimorphisms and derivations on F-algebras and some algebras of vector-valued functions are considered.
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Zhuraev, Tursunbay. "Some Properties of the Covariant Functor Set of Exponential Type." Revista Gestão Inovação e Tecnologias 11, no. 2 (June 5, 2021): 1139–52. http://dx.doi.org/10.47059/revistageintec.v11i2.1743.
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In this paper, it is shown that the sets of all non-empty subsets Set (x) of a topological space X with exponential topology is a covariant functor in the category of -topological spaces and their continuous mappings into itself. It is shown that the functor Set is a covariant functor in the category of topological spaces and continuous mappings into itself, a pseudometric in the space Set (x) is defined, and compact, connected, finite, and countable subspaces of Set (x) are distinguished. It also shows various kinds of connectivity, soft, locally soft, and n - soft mappings in Set (x). One interesting example is given for the TOPY category. It is proved that the functor Set maps open mappings to open, contractible and locally contractible spaces and into contractible and locally contractible spaces.
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Corbacho, E., V. Tarieladze, and R. Vidal. "Equicontinuity and Quasi-Uniformities." gmj 11, no. 4 (December 2004): 681–90. http://dx.doi.org/10.1515/gmj.2004.681.
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Abstract For topological spaces 𝑋, 𝑌 with a fixed compatible quasi-uniformity 𝑄 in 𝑌 and for a family (𝑓𝑖)𝑖∈𝐼 of mappings from 𝑋 to 𝑌, the notions of even continuity in the sense of Kelley, topological equicontinuity in the sense of Royden and 𝑄-equicontinuity (i.e., equicontinuity with respect to the topology of 𝑋 and 𝑄) are compared. It is shown that 𝑄-equicontinuity implies even continuity, and if 𝑄 is locally symmetric, it implies topological equicontinuity too. It turns out that these notions are equivalent provided 𝑄 is a uniformity compatible with a compact topology, but the equivalence may fail even for a locally symmetric quasi-uniformity 𝑄 compatible with a compact metrizable topology.
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Holá, Ľubica. "The Attouch-Wets topology and a characterisation of normable linear spaces." Bulletin of the Australian Mathematical Society 44, no. 1 (August 1991): 11–18. http://dx.doi.org/10.1017/s0004972700029415.
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Let X and Y be metric spaces and C(X, Y) be the space of all continuous functions from X to Y. If X is a locally connected space, the compact-open topology on C(X, Y) is weaker than the Attouch-Wets topology on C(X, Y). The result is applied on the space of continuous linear functions. Let X be a locally convex topological linear space metrisable with an invariant metric and X* be a continuous dual. X is normable if and only if the strong topology on X* and the Attouch-Wets topology coincide.
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Kukrak, Hleb O., and Vladimir L. Timokhovich. "On the continuity of functors of the type C(X, Y)." Journal of the Belarusian State University. Mathematics and Informatics, no. 1 (March 31, 2020): 22–29. http://dx.doi.org/10.33581/2520-6508-2020-1-22-29.
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We consider the category P, the objects of which are pairs of topological spaces (X, Y). Each such pair (X, Y) is assigned the space of continuous maps Cτ(X, Y) with some topology τ. By imposing some restrictions on objects and morphisms of category P, we define a subcategory K ⊂ P, for which the above map is a functor from K to the category Top of topological spaces and continuous maps. The following question is investigated. What are the additional conditions on K, under which the above functor is continuous? Along the way the problem of finding the limit of the inverse spectrum in the category P is solved. We show, that it reduces to finding the limits of the corresponding direct spectrum and inverse spectrum in the category Top. Point convergence topology, compact-open topology and graph topology are considered as the topology τ.
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(Papazyan), Talin Budak. "Compactifications of discrete versions of semitopological semigroups by filters of zero sets." Mathematical Proceedings of the Cambridge Philosophical Society 109, no. 2 (March 1991): 363–73. http://dx.doi.org/10.1017/s0305004100069826.
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AbstractThe maximal proper prime filters together with the ultrafilters of zero sets of any metrizable compact topological space are shown to have a compact Hausdorff topology in which the ultrafilters form a discrete, dense subspace. This gives a general theory of compactifications of discrete versions of compact metrizable topological spaces and some of the already known constructions of compact right topological semigroups are special cases of the general theory. In this way, simpler and more elegant proofs for these constructions are obtained.In [8], Pym constructed compactifications for discrete semigroups which can be densely embedded in a compact group. His techniques made extensive use of function algebras. In [4] Helmer and Isik obtained the same compactifications by using the existence of Stone ech compactifications. The aim of this paper is to present a general theory of compactifications of semitopological semigroups so that Helmer and Isik's results in [4] are a simple consequence. Our proofs are different and are based on filters which provide a natural way of getting compactifications. Moreover we present new insights by emphasizing maximal proper primes which are not ultrafilters.We start by defining filters of zero sets (called z-filters) on a given topological space X, and their convergence. In the case of compact metrizable topological spaces, we establish the connections between proper maximal prime z-filters on X and zultrafilters in β(X\{x})\(X\{x}) where β(X\{x}) is the Stone-ech compactification of X\{x}. We then define a topology on the set of all prime z-filters on X such that the subspace of all proper maximal primes is compact Hausdorff. We denote by the set of all proper maximal prime z-filters on X together with the z-ultrafilters and show that when X is a compact metrizable cancellative semitopological semigroup, is a compact right topological semigroup with dense topological centre. Also, when is considered for a compact Hausdorff metrizable group, the semigroup obtained is exactly the same (algebraically and topologically) as the semigroup obtained in [4]. Hence the result in [4] is just a consequence of the general theory presented in this paper.
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Howard, Paul E. "Definitions of compact." Journal of Symbolic Logic 55, no. 2 (June 1990): 645–55. http://dx.doi.org/10.2307/2274654.
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Several definitions of “compact” for topological spaces have appeared in the literature (see [5]). We will consider the following:Definition. A topological space X is1. Compact(1) if every open cover of X has a finite subcover.2. Compact(2) if every infinite subset E of X has a complete accumulation point (i.e., a point x0 ∈ X such that for every neighborhood U of x0, |E ∩ U| = |E|).3. Compact(3) if there is a subbase S for the topology on X such that every cover of X by members of S has a finite subcover.4. Compact(4) if each nest of closed, nonempty sets has a nonempty intersection.5. Compact(5) if every family of closed sets in X which has the finite intersection property (every finite subfamily has a nonempty intersection) has a nonempty intersection.6. Compact(6) if each net in X has a cluster point.7. Compact(7) if each net in X has a convergent subnet.This work was motivated primarily by consideration of various proofs that the Tychonoff theorem, T (“the product of compact topological spaces is compact”) is equivalent to the Axiom of Choice, AC. Tychonoff's original proof that AC implies T used Definition 2 [13]. Other proofs have used Definitions 3 and 5; see [5]. The proof by Kelley that T implies AC uses Definition 5 [6].
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MUÑOZ, MARÍA. "BOUNDING THE MINIMUM ORDER OF Aπ-BASE." Bulletin of the Australian Mathematical Society 83, no. 2 (February 10, 2011): 321–28. http://dx.doi.org/10.1017/s0004972710001966.
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AbstractLetXbe a topological space. A family ℬ of nonempty open sets inXis called aπ-base ofXif for each open setUinXthere existsB∈ℬ such thatB⊂U. The order of aπ-base ℬ at a pointxis the cardinality of the family ℬx={B∈ℬ:x∈B} and the order of theπ-base ℬ is the supremum of the orders of ℬ at each pointx∈X. A classical theorem of Shapirovskiĭ [‘Special types of embeddings in Tychonoff cubes’, in:Subspaces of Σ-Products and Cardinal Invariants, Topology, Coll. Math. Soc. J. Bolyai, 23 (North-Holland, Amsterdam, 1980), pp. 1055–1086; ‘Cardinal invariants in compact Hausdorff spaces’,Amer. Math. Soc. Transl.134(1987), 93–118] establishes that the minimum order of aπ-base is bounded by the tightness of the space when the space is compact. Since then, there have been many attempts at improving the result. Finally, in [‘The projectiveπ-character bounds the order of aπ-base’,Proc. Amer. Math. Soc.136(2008), 2979–2984], Juhász and Szentmiklóssy proved that the minimum order of aπ-base is bounded by the ‘projectiveπ-character’ of the space for any topological space (not only for compact spaces), improving Shapirovskiĭ’s theorem. The projectiveπ-character is in some sense an ‘external’ cardinal function. Our purpose in this paper is, on the one hand, to give bounds of the projectiveπ-character using ‘internal’ topological properties of the subspaces on compact spaces. On the other hand, we give a bound on the minimum order of aπ-base using other cardinal functions in the frame of general topological spaces. Open questions are posed.
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BEZHANISHVILI, NICK, and WESLEY H. HOLLIDAY. "CHOICE-FREE STONE DUALITY." Journal of Symbolic Logic 85, no. 1 (August 29, 2019): 109–48. http://dx.doi.org/10.1017/jsl.2019.11.
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AbstractThe standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras.
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Aguadé, J. "Decomposable Free Loop Spaces." Canadian Journal of Mathematics 39, no. 4 (August 1, 1987): 938–55. http://dx.doi.org/10.4153/cjm-1987-047-9.
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In this paper we study the spaces X having the property that the space of free loops on X is equivalent in some sense to the product of X by the space of based loops on X. We denote by ΛX the space of all continuous maps from S1 to X, with the compact-open topology. ΩX denotes, as usual, the loop space of X, i.e., the subspace of ΛX formed by the maps from S1 to X which map 1 to the base point of X.If G is a topological group then every loop on G can be translated to the base point of G and the space of free loops ΛG is homeomorphic to G × ΩG. More generally, any H-space has this property up to homotopy. Our purpose is to study from a homotopy point of view the spaces X for which there is a homotopy equivalence between ΛX and X × ΩX which is compatible with the inclusion ΩX ⊂ ΛX and the evaluation map ΛX → X.
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SCHRÖDER, MATTHIAS. "The sequential topology on is not regular." Mathematical Structures in Computer Science 19, no. 5 (September 8, 2009): 943–57. http://dx.doi.org/10.1017/s0960129509990065.
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The compact-open topology on the set of continuous functionals from the Baire space to the natural numbers is well known to be zero-dimensional. We prove that the closely related sequential topology on this set is not even regular. The sequential topology arises naturally as the topology carried by the exponential formed in various cartesian closed categories of topological spaces. Moreover, we give an example of an effectively open subset of that violates regularity. The topological properties of are known to be closely related to an open problem in Computable Analysis. We also show that the sequential topology on the space of continuous real-valued functions on a Polish space need not be regular.