ON THE SPECTRALIFICATION OF A HEMISPECTRAL SPACE

2011 ◽  
Vol 10 (04) ◽  
pp. 687-699
Author(s):  
OTHMAN ECHI ◽  
MOHAMED OUELD ABDALLAHI
Keyword(s):  
Topological Space ◽  
Open Subset ◽  
Full Subcategory ◽  
Topological Spaces ◽  
Open Problems ◽  
Complete Answer ◽  
Reflective Subcategory ◽  
Continuous Map ◽  
Open Set ◽  
Spectral Spaces

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."

Degrees of recursively enumerable topological spaces

1983 ◽  
Vol 48 (3) ◽  
pp. 610-622 ◽  
Author(s):  
Iraj Kalantari ◽  
J. B. Remmel
Keyword(s):  
Topological Space ◽  
Open Subset ◽  
The Body ◽  
Topological Spaces ◽  
Open Set ◽  
Recursively Enumerable ◽  
Closed Fields ◽  
Algebraically Closed Fields ◽  
Nowhere Dense Set ◽  
Dense Set

In [5], Metakides and Nerode introduced the study of recursively enumerable (r.e.) substructures of a recursively presented structure. The main line of study presented in [5] is to examine the effective content of certain algebraic structures. In [6], Metakides and Nerode studied the lattice of r.e. subspaces of a recursively presented vector space. This lattice was later studied by Kalantari, Remmel, Retzlaff and Shore. Similar studies have been done by Metakides and Nerode [7] for algebraically closed fields, by Remmel [10] for Boolean algebras and by Metakides and Remmel [8] and [9] for orderings. Kalantari and Retzlaff [4] introduced and studied the lattice of r.e. subsets of a recursively presented topological space. Kalantari and Retzlaff consideredX, a topological space with ⊿, a countable basis. This basis is coded into integers and with the help of this coding, r.e. subsets ofωgive rise to r.e. subsets ofX. The notion of “recursiveness” of a topological space is the natural next step which gives rise to the question of what should be the “degree” of an r.e. open subset ofX? It turns out that any r.e. open set partitions ⊿; into four sets whose Turing degrees become central in answering the question raised above.In this paper we show that the degrees of the elements of the partition of ⊿ imposed by an r.e. open set can be “controlled independently” in a sense to be made precise in the body of the paper. In [4], Kalantari and Retzlaff showed that givenAany r.e. set andany r.e. open subset ofX, there exists an r.e. open set ℋ which is a subset ofand is dense in(in a topological sense) and in whichAis coded. This shows that modulo a nowhere dense set, an r.e. open set can become as complicated as desired. After giving the general technical and notational machinery in §1, and giving the particulars of our needs in §2, in §3 we prove that the set ℋ described above could be made to be precisely of degree ofA. We then go on and establish various results (both existential and universal) on the mentioned partitioning of ⊿. One of the surprising results is that there are r.e. open sets such that every element of partitioning of ⊿ is of a different degree. Since the exact wording of the results uses the technical definitions of these partitioning elements, we do not summarize the results here and ask the reader to examine §3 after browsing through §§1 and 2.

scholarly journals On Some R1-Properties in Fuzzy Topological Spaces

2011 ◽  
Vol 4 (1) ◽  
pp. 21
Author(s):  
D. M. Ali ◽  
F. A. Azam
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Complete Answer ◽  
Fuzzy Topological Space ◽  
Fuzzy Topological Spaces

In this paper, we introduce six R1-axioms for fuzzy topological spaces (in short, fts). We study their interrelations, goodness and initialities. Besides we recall nine R0-axioms for fts. A complete answer is given with regard to all possible (R1=>R0)-type implications for fts.Keywords: Fuzzy topological space; Fuzzy R1-axiom; Fuzzy R0-axiom.© 2012 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.doi:10.3329/jsr.v4i1.7044               J. Sci. Res. 4 (1), 21-32  (2012) 

A View on Fuzzy Minimal Open Sets and Fuzzy Maximal Open Sets

2015 ◽  
Vol 7 (1) ◽  
pp. 62-73 ◽  
Author(s):  
Hamid Reza Moradi
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Basic Properties ◽  
New Class ◽  
Properties And Characterization ◽  
Open Set ◽  
Fuzzy Topological Space ◽  
Fuzzy Topological Spaces ◽  
Characterization Theorems

A nonzero fuzzy open set () of a fuzzy topological space is said to be fuzzy minimal open (resp. fuzzy maximal open) set if any fuzzy open set which is contained (resp. contains) in is either or itself (resp. either or itself). In this note, a new class of sets called fuzzy minimal open sets and fuzzy maximal open sets in fuzzy topological spaces are introduced and studied which are subclasses of open sets. Some basic properties and characterization theorems are also to be investigated.

scholarly journals Constructing Banaschewski compactification without Dedekind completeness axiom

2004 ◽  
Vol 2004 (69) ◽  
pp. 3799-3816
Author(s):  
S. K. Acharyya ◽  
K. C. Chattopadhyay ◽  
Partha Pratim Ghosh
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Open Problems ◽  
Structure Space ◽  
Ordered Field ◽  
Completeness Axiom ◽  
Dedekind Completeness ◽  
Stone Theorem ◽  
Stone’S Theorem

The main aim of this paper is to provide a construction of the Banaschewski compactification of a zero-dimensional Hausdorff topological space as a structure space of a ring of ordered field-valued continuous functions on the space, and thereby exhibit the independence of the construction from any completeness axiom for an ordered field. In the process of describing this construction we have generalized the classical versions of M. H. Stone's theorem, the Banach-Stone theorem, and the Gelfand-Kolmogoroff theorem. The paper is concluded with a conjecture of a split in the class of all zero-dimensional but not strongly zero-dimensional Hausdorff topological spaces into three classes that are labeled by inequalities between three compactifications ofX, namely, the Stone-Čech compactificationβX, the Banaschewski compactificationβ0X, and the structure space𝔐X,Fof the lattice-ordered commutative ringℭ(X,F)of all continuous functions onXtaking values in the ordered fieldF, equipped with its order topology. Some open problems are also stated.

scholarly journals On β-Open Sets and Ideals in Topological Spaces

2019 ◽  
Vol 12 (3) ◽  
pp. 893-905
Author(s):  
Glaisa T. Catalan ◽  
Roberto N. Padua ◽  
Michael Jr. Patula Baldado
Keyword(s):  
Topological Space ◽  
Compact Space ◽  
Topological Spaces ◽  
Open Set ◽  
Separated Sets ◽  
The Ideal

Let X be a topological space and I be an ideal in X. A subset A of a topological space X is called a β-open set if A ⊆ cl(int(cl(A))). A subset A of X is called β-open with respect to the ideal I, or βI -open, if there exists an open set U such that (1) U − A ∈ I, and (2) A − cl(int(cl(U))) ∈ I. A space X is said to be a βI -compact space if it is βI -compact as a subset. An ideal topological space (X, τ, I) is said to be a cβI -compact space if it is cβI -compact as a subset. An ideal topological space (X, τ, I) is said to be a countably βI -compact space if X is countably βI -compact as a subset. Two sets A and B in an ideal topological space (X, τ, I) is said to be βI -separated if clβI (A) ∩ B = ∅ = A ∩ clβ(B). A subset A of an ideal topological space (X, τ, I) is said to be βI -connected if it cannot be expressed as a union of two βI -separated sets. An ideal topological space (X, τ, I) is said to be βI -connected if X βI -connected as a subset. In this study, we introduced the notions βI -open set, βI -compact, cβI -compact, βI -hyperconnected, cβI -hyperconnected, βI -connected and βI -separated. Moreover, we investigated the concept β-open set by determining some of its properties relative to the above-mentioned notions.

scholarly journals Relations of Pre Generalized Regular Weakly Locally Closed Sets in Topological Spaces

2021 ◽  
Vol 12 (3) ◽  
pp. 3444-3454
Author(s):  
Vijayakumari T Et.al
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Closed Set ◽  
Closed Sets ◽  
Open Set ◽  
Continuous Maps ◽  
Locally Closed Sets

In this paper pgrw-locally closed set, pgrw-locally closed*-set and pgrw-locally closed**-set are introduced. A subset A of a topological space (X,t) is called pgrw-locally closed (pgrw-lc) if A=GÇF where G is a pgrw-open set and F is a pgrw-closed set in (X,t). A subset A of a topological space (X,t) is a pgrw-lc* set if there exist a pgrw-open set G and a closed set F in X such that A= GÇF. A subset A of a topological space (X,t) is a pgrw-lc**-set if there exists an open set G and a pgrw-closed set F such that A=GÇF. The results regarding pgrw-locally closed sets, pgrw-locally closed* sets, pgrw-locally closed** sets, pgrw-lc-continuous maps and pgrw-lc-irresolute maps and some of the properties of these sets and their relation with other lc-sets are established.

scholarly journals Neutrosophic Orbit Topological Spaces

2021 ◽  
Vol 18 (24) ◽  
pp. 1443
Author(s):  
T Madhumathi ◽  
F NirmalaIrudayam
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Topological Space ◽  
Necessary Conditions ◽  
Neutrosophic Set ◽  
Topological Spaces ◽  
Open Set ◽  
Evolution Function

Neutrosophy is a flourishing arena which conceptualizes the notion of true, falsity and indeterminancy attributes of an event. In the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. Hence in this paper we focus on introducing the concept of neutrosophic orbit topological space denoted as (X, tNO). Also, some of the important characteristics of neutrosophic orbit open sets are discussed with suitable examples. HIGHLIGHTS The orbit in mathematics has an important role in the study of dynamical systems Neutrosophy is a flourishing arena which conceptualizes the notion of true, falsity and indeterminancy attributes of an event. We combine the above two topics and create the following new concept The collection of all neutrosophic orbit open sets under the mapping . we introduce the necessary conditions on the mapping 𝒇 in order to obtain a fixed orbit of a neutrosophic set (i.e., 𝒇(𝝁) = 𝝁) for any neutrosophic orbit open set 𝝁 under the mapping 𝒇

Locally Convex Hypersurfaces

1973 ◽  
Vol 25 (3) ◽  
pp. 531-538 ◽  
Author(s):  
L. B. Jonker ◽  
R. D. Norman
Keyword(s):  
Convex Body ◽  
Open Subset ◽  
Convex Hypersurface ◽  
Topological Manifold ◽  
Continuous Map ◽  
Open Set ◽  
Locally Convex ◽  
Convex Hypersurfaces

Let M be an n-dimensional connected topological manifold. Let ξ : M → Rn+1 be a continuous map with the following property: to each x ∈ M there is an open set x ∈ Ux ⊂ M, and a convex body Kx ⊂ Rn+1 such that ξ(UX) is an open subset of ∂Kx and such that is a homeomorphism onto its image. We shall call such a mapping ξ a locally convex immersion and, along with Van Heijenoort [8] we shall call ξ(M) a locally convex hypersurface of Rn+1.

scholarly journals Soft ideal topological space and mixed fuzzy soft ideal topological space

2017 ◽  
Vol 37 (1) ◽  
pp. 141-151
Author(s):  
Manash Borah ◽  
Bipan Hazarika
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Open Set

In this paper we introduce fuzzy soft ideal and mixed fuzzy soft ideal topological spaces and some properties of this space. Also we introduce fuzzy soft $I$-open set, fuzzy soft $\alpha$-$I$-open set, fuzzy soft pre-$I$-open set, fuzzy soft semi-$I$-open set and fuzzy soft $\beta$-$I$-open set and discuss some of their properties.

scholarly journals Pre Separation Axioms In Neutrosophic Crisp Topological Spaces

2020 ◽  
pp. 72-79
Author(s):  
Riad K. Al Al-Hamido ◽  
◽  
◽  
◽  
Luai Salha ◽  
...  
Keyword(s):  
Topological Space ◽  
The Other ◽  
Topological Spaces ◽  
Open Set ◽  
Separation Axioms ◽  
New Type

In this paper, A new type of separation axioms in the neutrosophic crisp Topological space named neutrosophic crisp pre separation axioms is going to be defined , in which neutrosophic crisp pre open set and neutrosophic crisp point are to be depended on. Also, relations among them and the other type are going to be found.

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