scholarly journals EXTENSION OF CONTINUOUS MAPPINGS AND H1-RETRACTS

2008 ◽  
Vol 78 (3) ◽  
pp. 497-506 ◽  
Author(s):  
OLENA KARLOVA
Keyword(s):  
Topological Space ◽  
Continuous Mapping ◽  
Paracompact Space ◽  
Open Set ◽  
Continuous Mappings ◽  
Arbitrary Topological Space ◽  
Completely Metrizable

AbstractWe prove that any continuous mapping f:E→Y on a completely metrizable subspace E of a perfect paracompact space X can be extended to a Lebesgue class one mapping g:X→Y (that is, for every open set V in Y the preimage g−1(V ) is an Fσ-set in X) with values in an arbitrary topological space Y.

scholarly journals On Strongly b − θ -Continuous Mappings in Fuzzifying Topology

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Ting Yang ◽  
Ahmed Mostafa Khalil
Keyword(s):  
Fuzzy Logic ◽  
Continuous Mapping ◽  
Topological Spaces ◽  
Closed Set ◽  
Open Set ◽  
Neighborhood System ◽  
Continuous Mappings ◽  
Derived Set ◽  
New Type

In this article, we will define the new notions (e.g., b − θ -neighborhood system of point, b − θ -closure (interior) of a set, and b − θ -closed (open) set) based on fuzzy logic (i.e., fuzzifying topology). Then, we will explain the interesting properties of the above five notions in detail. Several basic results (for instance, Definition 7, Theorem 3 (iii), (v), and (vi), Theorem 5, Theorem 9, and Theorem 4.6) in classical topology are generalized in fuzzy logic. In addition to, we will show that every fuzzifying b − θ -closed set is fuzzifying γ -closed set (by Theorem 3 (vi)). Further, we will study the notion of fuzzifying b − θ -derived set and fuzzifying b − θ -boundary set and discuss several of their fundamental basic relations and properties. Also, we will present a new type of fuzzifying strongly b − θ -continuous mapping between two fuzzifying topological spaces. Finally, several characterizations of fuzzifying strongly b − θ -continuous mapping, fuzzifying strongly b − θ -irresolute mapping, and fuzzifying weakly b − θ -irresolute mapping along with different conditions for their existence are obtained.

scholarly journals Compactness in intuitionistic fuzzy topological spaces

2005 ◽  
Vol 2005 (1) ◽  
pp. 19-32 ◽  
Author(s):  
A. A. Ramadan ◽  
S. E. Abbas ◽  
A. A. Abd El-Latif
Keyword(s):  
Topological Space ◽  
Continuous Mapping ◽  
Topological Spaces ◽  
Intuitionistic Fuzzy ◽  
Continuous Mappings ◽  
Weakly Continuous ◽  
Fuzzy Topological Space ◽  
Definition Of ◽  
Fuzzy Topological Spaces ◽  
Almost Continuous

We introduce fuzzy almost continuous mapping, fuzzy weakly continuous mapping, fuzzy compactness, fuzzy almost compactness, and fuzzy near compactness in intuitionistic fuzzy topological space in view of the definition of Šostak, and study some of their properties. Also, we investigate the behavior of fuzzy compactness under several types of fuzzy continuous mappings.

scholarly journals Fuzzy Almost Generalized E-Continuous Mappings in Smooth Topological Spaces

2019 ◽  
Vol 8 (10S) ◽  
pp. 138-141 ◽  
Keyword(s):  
Topological Space ◽  
Compact Space ◽  
Continuous Mapping ◽  
Regular Space ◽  
Topological Spaces ◽  
Continuous Mappings ◽  
Fuzzy Topological Space ◽  
Index Terms ◽  
Almost Continuous

We introduce and study several interesting properties of fuzzy almost generalized e -continuous mappings in smooth topological spaces with counter examples. We also introduce fuzzy r - 1 2 fT e -space, r -fuzzy ge -space, r -fuzzy regular ge -space and r -fuzzy generalized e -compact space. It is seen that a fuzzy almost generalized e -continuous mapping, between a fuzzy r - 1 2 fT e -space and a fuzzy topological space, becomes fuzzy almost continuous mapping. Index Terms: fage -continuous, r - fge -space, r - fge -regular space, r - 1 2 fT e -space.

Perturbation of the fixed point problem for continuous mappings

2020 ◽  
pp. 241-249
Author(s):  
Zukhra T. Zhukovskaya ◽  
Sergey E. Zhukovskiy
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Continuous Mapping ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Bounded Subset ◽  
Completely Continuous ◽  
Continuous Map ◽  
Continuous Mappings

We consider the problem of a double fixed point of pairs of continuous mappings defined on a convex closed bounded subset of a Banach space. It is shown that if one of the mappings is completely continuous and the other is continuous, then the property of the existence of fixed points is stable under contracting perturbations of the mappings. We obtain estimates for the distance from a given pair of points to double fixed points of perturbed mappings. We consider the problem of a fixed point of a completely continuous mapping on a convex closed bounded subset of a Banach space. It is shown that the property of the existence of a fixed point of a completely continuous map is stable under contracting perturbations. Estimates of the distance from a given point to a fixed point are obtained. As an application of the obtained results, the solvability of a difference equation of a special type is proved.

scholarly journals On Soft Rough Topology with Multi-Attribute Group Decision Making

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 67 ◽  
Author(s):  
Muhammad Riaz ◽  
Florentin Smarandache ◽  
Atiqa Firdous ◽  
Atiqa Fakhar
Keyword(s):  
Decision Making ◽  
Rough Set ◽  
Group Decision Making ◽  
Topological Structure ◽  
Group Decision ◽  
Continuous Mapping ◽  
Product Topology ◽  
Closed Sets ◽  
Open Set ◽  
Soft Rough Set

Rough set approaches encounter uncertainty by means of boundary regions instead of membership values. In this paper, we develop the topological structure on soft rough set ( SR -set) by using pairwise SR -approximations. We define SR -open set, SR -closed sets, SR -closure, SR -interior, SR -neighborhood, SR -bases, product topology on SR -sets, continuous mapping, and compactness in soft rough topological space ( SRTS ). The developments of the theory on SR -set and SR -topology exhibit not only an important theoretical value but also represent significant applications of SR -sets. We applied an algorithm based on SR -set to multi-attribute group decision making (MAGDM) to deal with uncertainty.

scholarly journals On s-g-cocompact open set and Continuity

2020 ◽  
Vol 25 (2) ◽  
pp. 67-77 ◽  
Author(s):  
Raad Al-Abdulla ◽  
Salam Jabar
Keyword(s):  
Topological Space ◽  
Open Set ◽  
The Relationship

    Throughout this paper by a space we mean a supra topological space, we have studied some of propertiese to new set is called supra generalize- cocompact open set ( -g-coc-open set)and find the relation with other sets and our concluded anew class of the function called -g-coc-continuous, -g-coc'-continuous, -coc-continuous, -coc'-continuous We shall provided some properties of these concepts and it will explain the relationship among them and some results on this subjects are proved Throughout this work , and new concept have been illustrated including , -coc-ompact space .

scholarly journals On B-Open Sets

2019 ◽  
Vol 12 (2) ◽  
pp. 358-369
Author(s):  
Layth Muhsin Habeeb Alabdulsada
Keyword(s):  
Topological Space ◽  
Open Set

The aim of this paper is to introduce and study $\mathcal{B}$-open sets and related properties. Also, we define a bi-operator topological space $(X, \tau, T_1, T_2)$, involving the two operators $T_1$ and $T_2$, which are used to define $\mathcal{B}$-open sets. A $\mathcal{B}$-open set is, roughly speaking, a generalization of a $b$-open set, which is, in turn, a generalization of a pre-open set and a semi-open set. We introduce a number of concepts based on $\mathcal{B}$-open sets.

scholarly journals Paracompactness with respect to an ideal

1997 ◽  
Vol 20 (3) ◽  
pp. 433-442 ◽  
Author(s):  
T. R. Hamlett ◽  
David Rose ◽  
Dragan Janković
Keyword(s):  
Topological Space ◽  
Finite Union ◽  
Open Cover ◽  
Locally Finite ◽  
Open Set ◽  
Special Cases ◽  
Paracompact Spaces

An ideal on a setXis a nonempty collection of subsets ofXclosed under the operations of subset and finite union. Given a topological spaceXand an idealℐof subsets ofX,Xis defined to beℐ-paracompact if every open cover of the space admits a locally finite open refinement which is a cover for all ofXexcept for a set inℐ. Basic results are investigated, particularly with regard to theℐ- paracompactness of two associated topologies generated by sets of the formU−IwhereUis open andI∈ℐand⋃{U|Uis open andU−A∈ℐ, for some open setA}. Preservation ofℐ-paracompactness by functions, subsets, and products is investigated. Important special cases ofℐ-paracompact spaces are the usual paracompact spaces and the almost paracompact spaces of Singal and Arya [“On m-paracompact spaces”, Math. Ann., 181 (1969), 119-133].

scholarly journals Lebesgue Measurability of Separately Continuous Functions and Separability

2007 ◽  
Vol 2007 ◽  
pp. 1-4
Author(s):  
V. V. Mykhaylyuk
Keyword(s):  
Topological Space ◽  
Chain Condition ◽  
Continuous Functions ◽  
Negative Answer ◽  
Regular Space ◽  
Property A ◽  
Completely Regular ◽  
Open Set ◽  
Countable Chain Condition ◽  
Countable Chain

A connection between the separability and the countable chain condition of spaces withL-property (a topological spaceXhasL-property if for every topological spaceY, separately continuous functionf:X×Y→ℝand open setI⊆ℝ,the setf−1(I)is anFσ-set) is studied. We show that every completely regular Baire space with theL-property and the countable chain condition is separable and constructs a nonseparable completely regular space with theL-property and the countable chain condition. This gives a negative answer to a question of M. Burke.

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