A note on λ-compact spaces

2013 ◽  
Vol 63 (6) ◽  
Author(s):  
O. Karamzadeh ◽  
M. Namdari ◽  
M. Siavoshi
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Compact Spaces

AbstractWe extend the well-known and important fact that “a topological space X is compact if and only if every ideal in C(X) is fixed”, to more general topological spaces. Some interesting consequences are also observed. In particular, the maximality of compact Hausdorff spaces with respect to the property of compactness is generalized and the topological spaces with this generalized property are characterized.

scholarly journals BOUNDING THE MINIMUM ORDER OF Aπ-BASE

2011 ◽  
Vol 83 (2) ◽  
pp. 321-328
Author(s):  
MARÍA MUÑOZ
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Minimum Order ◽  
Compact Spaces ◽  
Cardinal Invariants ◽  
Open Set ◽  
Open Questions ◽  
The Family ◽  
The One

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.

Essentials of General Topology

2021 ◽  
pp. 191-244
Author(s):  
Adel N. Boules
Keyword(s):  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Compact Spaces ◽  
Section 8 ◽  
Finite Products ◽  
The One ◽  
One Point Compactification ◽  
Locally Compact Spaces ◽  
Metrization Theorem

The first eight sections of this chapter constitute its core and are generally parallel to the leading sections of chapter 4. Most of the sections are brief and emphasize the nonmetric aspects of topology. Among the topics treated are normality, regularity, and second countability. The proof of Tychonoff’s theorem for finite products appears in section 8. The section on locally compact spaces is the transition between the core of the chapter and the more advanced sections on metrization, compactification, and the product of infinitely many spaces. The highlights include the one-point compactification, the Urysohn metrization theorem, and Tychonoff’s theorem. Little subsequent material is based on the last three sections. At various points in the book, it is explained how results stated for the metric case can be extended to topological spaces, especially locally compact Hausdorff spaces. Some such results are developed in the exercises.

scholarly journals Compact and extremally disconnected spaces

2004 ◽  
Vol 2004 (20) ◽  
pp. 1047-1056
Author(s):  
Bhamini M. P. Nayar
Keyword(s):  
Topological Space ◽  
Compact Space ◽  
Closed Subset ◽  
Hausdorff Space ◽  
Hausdorff Spaces ◽  
Compact Spaces ◽  
Hausdorff Topological Space ◽  
Extremally Disconnected

Viglino defined a Hausdorff topological space to beC-compact if each closed subset of the space is anH-set in the sense of Veličko. In this paper, we study the class of Hausdorff spaces characterized by the property that each closed subset is anS-set in the sense of Dickman and Krystock. Such spaces are calledC-s-compact. Recently, the notion of strongly subclosed relation, introduced by Joseph, has been utilized to characterizeC-compact spaces as those with the property that each function from the space to a Hausdorff space with a strongly subclosed inverse is closed. Here, it is shown thatC-s-compact spaces are characterized by the property that each function from the space to a Hausdorff space with a strongly sub-semiclosed inverse is a closed function. It is established that this class of spaces is the same as the class of Hausdorff, compact, and extremally disconnected spaces. The class ofC-s-compact spaces is properly contained in the class ofC-compact spaces as well as in the class ofS-closed spaces of Thompson. In general, a compact space need not beC-s-compact. The product of twoC-s-compact spaces need not beC-s-compact.

scholarly journals Applications on operations on weakly compact generalized topological spaces

2020 ◽  
Vol 12 (2) ◽  
pp. 461-467
Author(s):  
B. Roy ◽  
T. Noiri
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Weakly Compact ◽  
Compact Spaces ◽  
Generalized Topological Space ◽  
Compact Subsets

In this paper, we have introduced the notion of operations on a generalized topological space $(X,\mu)$ to investigate the notion of $\gamma_{_\mu}$-compact subsets of a generalized topological space and to study some of its properties. It is also shown that, under some conditions, $\gamma_{_\mu}$-compactness of a space is equivalent to some other weak forms of compactness. Characterizations of such sets are given. We have then introduced the concept of $\gamma_{_\mu}$-$T_{_2}$ spaces to study some properties of $\gamma_{_\mu}$-compact spaces. This operation enables us to unify different results due to S. Kasahara.

Locally Compact Normal Spaces in the Constructible Universe

1982 ◽  
Vol 34 (5) ◽  
pp. 1091-1096 ◽  
Author(s):  
W. Stephen Watson
Keyword(s):  
Set Theory ◽  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Compact Spaces ◽  
Locally Compact Spaces ◽  
Perfectly Normal

Arhangel'skiĭ proved around 1959 [1] that, for the class of perfectly normal locally compact spaces, metacompactness and paracompactness are equivalent. It is shown to be consistent that this equivalence holds for the (larger) class of normal locally compact spaces (answering a question of Tall [8], [9]).The consistency of the existence of locally compact normal noncollectionwise Hausdorff spaces has been known since 1967. It is shown that the existence of such spaces is independent of the axioms of set theory, thus establishing that Bing's example G cannot be modified under ZFC to be locally compact.All topological spaces are assumed to be Hausdorff.First, a definition and three standard lemmata are needed.

scholarly journals α-fuzzy compactness inI-topological spaces

2003 ◽  
Vol 2003 (41) ◽  
pp. 2609-2617
Author(s):  
Valentín Gregori ◽  
Hans-peter A. Künzi
Keyword(s):  
Topological Space ◽  
Baire Category ◽  
Unit Interval ◽  
Topological Spaces ◽  
Baire Category Theorem ◽  
Locally Compact ◽  
Compact Spaces ◽  
Category Theorem ◽  
Locally Compact Spaces

Using a gradation of openness in a (Chang fuzzy)I-topological space, we introduce degrees of compactness that we callα-fuzzy compactness (whereαbelongs to the unit interval), so extending the concept of compactness due to C. L. Chang. We obtain a Baire category theorem forα-locally compact spaces and construct a one-pointα-fuzzy compactification of anI-topological space.

Continuous images of proper analytic and proper Borel spaces

1974 ◽  
Vol 75 (2) ◽  
pp. 185-191 ◽  
Author(s):  
J. E. Jayne
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Completely Regular ◽  
Continuous Images

All topological spaces considered will be completely regular Hausdorff spaces. The word space will be used to refer to such a topological space.

Nearrings of Continuous Functions From Topological Spaces into Topological Nearrings

1996 ◽  
Vol 39 (3) ◽  
pp. 316-329
Author(s):  
K. D. Magill
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Hausdorff Spaces ◽  
Real Numbers

AbstractLet λ be a map from the additive Euclidean n-group Rn into the space R of real numbers and define a multiplication * on Rn by v * w = (λ(w))v. Then (Rn, + , *) is a topological nearring if and only if λ is continuous and λ(av) = aλ(v) for every v € Rn and every a in the range of λ. For any such map λ and any topological space X we denote by Nλ(X, Rn) the nearring of all continuous functions from X into (Rn, +, *) where the operations are pointwise. The ideals of Nλ(X, Rn) are investigated in some detail for certain λ and the results obtained are used to prove that two compact Hausdorff spaces X and Y are homeomorphic if and only if the nearrings Nλ(X, Rn) and Nλ(Y, Rn) are isomorphic.

scholarly journals Partial soft separation axioms and soft compact spaces

Filomat ◽  
2018 ◽  
Vol 32 (13) ◽  
pp. 4755-4771 ◽  
Author(s):  
M.E. El-Shafei ◽  
M. Abo-Elhamayel ◽  
T.M. Al-Shami
Keyword(s):  
Topological Space ◽  
General Topology ◽  
Regular Space ◽  
Topological Spaces ◽  
Finite Product ◽  
Compact Spaces ◽  
Separation Axioms ◽  
Soft Topological Space

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.

scholarly journals ON LOCAL WEAK τ-DENSITY OF TOPOLOGICAL SPACES

2021 ◽  
pp. 16-23
Author(s):  
F.G. Mukhamadiev ◽  
◽  
Keyword(s):  
Topological Space ◽  
Local Density ◽  
Cardinal Number ◽  
Sufficient Conditions ◽  
Topological Spaces ◽  
Locally Compact ◽  
Compact Spaces ◽  
Weak Separability ◽  
Weak Density ◽  
Locally Compact Spaces

A topological space X is locally weakly separable [3] at a point x∈X if x has a weakly separable neighbourhood. A topological space X is locally weakly separable if X is locally weakly separable at every point x∈X. The notion of local weak separability can be generalized for any cardinal τ ≥ℵ0 . A topological space X is locally weakly τ-dense at a point x∈X if τ is the smallest cardinal number such that x has a weak τ-dense neighborhood in X [4]. The local weak density at a point x is denoted as lwd(x). The local weak density of a topological space X is defined in following way: lwd ( X ) = sup{ lwd ( x) : x∈ X } . A topological space X is locally τ-dense at a point x∈X if τ is the smallest cardinal number such that x has a τ-dense neighborhood in X [4]. The local density at a point x is denoted as ld(x). The local density of a topological space X is defined in following way: ld ( X ) = sup{ ld ( x) : x∈ X } . It is known that for any topological space we have ld(X ) ≤ d(X ) . In this paper, we study questions of the local weak τ-density of topological spaces and establish sufficient conditions for the preservation of the property of a local weak τ-density of subsets of topological spaces. It is proved that a subset of a locally τ-dense space is also locally weakly τ-dense if it satisfies at least one of the following conditions: (a) the subset is open in the space; (b) the subset is everywhere dense in space; (c) the subset is canonically closed in space. A proof is given that the sum, intersection, and product of locally weakly τ-dense spaces are also locally weakly τ-dense spaces. And also questions of local τ-density and local weak τ-density are considered in locally compact spaces. It is proved that these two concepts coincide in locally compact spaces.

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