scholarly journals F-Rings of Continuous Functions I

1959 ◽  
Vol 11 ◽  
pp. 80-86 ◽  
Author(s):  
Barron Brainerd
Keyword(s):  
Topological Space ◽  
Banach Algebra ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
The Other ◽  
Topological Spaces ◽  
Closed Sets ◽  
Real Complex

It is well known (2, 4) that the ring of all real (complex) continuous functions on a compact Hausdorff space can be characterized algebraically as a Banach algebra which satisfies certain additional intrinsic conditions. It might be expected that rings of all continuous functions on other topological spaces also have algebraic characterizations. The main purpose of this note is to discuss two such characterizations. In both cases the characterizations are given in the terms of the theory of F-brings (1). In one case a characterization is given for the ring of all (real) continuous functions on a generalized P-space, that is, a zero-dimensional topological space in which the class of open-closed sets forms a σ-algebra. A Hausdorff generalized P-space is a P-space in the terminology of (3). In the other case a theorem of Sikorski (6) is employed to give a characterization of the ring of all (real) continuous functions on an upper X1-compact P-space.

scholarly journals Fine Fuzzy sp Closed Sets in Fine Fuzzy Topological Spaces

2020 ◽  
Vol 9 (3) ◽  
pp. 1306-1313
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Inverse Image ◽  
Continuous Functions ◽  
Topological Spaces ◽  
The Novel ◽  
Closed Set ◽  
Closed Sets ◽  
Fuzzy Topological Space ◽  
Fuzzy Topological Spaces

The main view of this article is the extended version of the fine topological space to the novel kind of space say fine fuzzy topological space which is developed by the notion called collection of quasi coincident of fuzzy sets. In this connection, fine fuzzy closed sets are introduced and studied some features on it. Further, the relationship between fine fuzzy closed sets with certain types of fine fuzzy closed sets are investigated and their converses need not be true are elucidated with necessary examples. Fine fuzzy continuous function is defined as the inverse image of fine fuzzy closed set is fine fuzzy closed and its interrelations with other types of fine fuzzy continuous functions are obtained. The reverse implication need not be true is proven with examples. Finally, the applications of fine fuzzy continuous function are explained by using the composition.

Functionals on Real C(S)

1978 ◽  
Vol 30 (03) ◽  
pp. 490-498 ◽  
Author(s):  
Nicholas Farnum ◽  
Robert Whitley
Keyword(s):  
Banach Algebra ◽  
Compact Hausdorff Space ◽  
Linear Functional ◽  
Hausdorff Space ◽  
Function Algebra ◽  
Continuous Functions ◽  
Codimension One ◽  
Maximal Ideals ◽  
Invertible Elements ◽  
Complex Valued

The maximal ideals in a commutative Banach algebra with identity have been elegantly characterized [5; 6] as those subspaces of codimension one which do not contain invertible elements. Also, see [1]. For a function algebra A, a closed separating subalgebra with constants of the algebra of complex-valued continuous functions on the spectrum of A, a compact Hausdorff space, this characterization can be restated: Let F be a linear functional on A with the property: (*) For each ƒ in A there is a point s, which may depend on f, for which F(f) = f(s).

Characterization of Fuzzy δg*-Closed Sets in Fuzzy Topological Spaces

2016 ◽  
Vol 5 (2) ◽  
pp. 1-12
Author(s):  
Anahid Kamali ◽  
Hamid Reza Moradi
Keyword(s):  
Topological Space ◽  
Fuzzy Sets ◽  
Topological Spaces ◽  
Basic Properties ◽  
Closed Sets ◽  
Research Article ◽  
Classical Paper ◽  
Fuzzy Topological Space ◽  
Fuzzy Topological Spaces

The purpose of this research article is to explain the meaning of g-closed sets in fuzzy topological spaces, which is more understandable to the readers and we find some of its basic properties. The concept of fuzzy sets was introduced by Zadeh in his classical paper (1965). Thereafter many investigations have been carried out, in the general theoretical field and also in different applied areas, based on this concept. The idea of fuzzy topological space was introduced by Chang (1968). The idea is more or less a generalization of ordinary topological spaces. Different aspects of such spaces have been developed, by several investigators. This paper is also devoted to the development of the theory of fuzzy topological spaces.

scholarly journals Gelfand theorem implies Stone representation theorem of Boolean rings

1995 ◽  
Vol 18 (4) ◽  
pp. 701-704
Author(s):  
Parfeny P. Saworotnow
Keyword(s):  
Boolean Algebra ◽  
Topological Space ◽  
Representation Theorem ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Boolean Rings ◽  
Stone Representation ◽  
Stone Theorem

Stone Theorem about representing a Boolean algebra in terms of open-closed subsets of a topological space is a consequence of the Gelfand Theorem about representing aB∗- algebra as the algebra of continuous functions on a compact Hausdorff space.

scholarly journals OPERATORS ON BANACH ALGEBRA VALUED FUNCTION SPACES

1992 ◽  
Vol 23 (3) ◽  
pp. 233-238
Author(s):  
JOR-TING CHAN
Keyword(s):  
Banach Algebra ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Function Spaces ◽  
Linear Operators ◽  
Locally Compact ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Locally Compact Hausdorff Space

Let $S$ be a locally compact Hausdorff space and let $A$ be a Banach algebra. Denote by $C_0(S, A)$ the Banach algebra of all $A$-valued continuous functions vanishing at infinity on $S$. Properties of bounded linear operators on $C_0(S,A)$, like multiplicativity, are characterized by Choy in terms of their representing measures. We study these theorems and give sharper results in certain cases.

DITKIN CONDITIONS

2017 ◽  
Vol 60 (1) ◽  
pp. 153-163
Author(s):  
AZADEH NIKOU ◽  
ANTHONY G. O'FARRELL
Keyword(s):  
Banach Algebra ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Commutative Banach Algebra ◽  
Algebraic Properties ◽  
The Relationship

AbstractThis paper is about the connection between certain Banach-algebraic properties of a commutative Banach algebra E with unit and the associated commutative Banach algebra C(X, E) of all continuous functions from a compact Hausdorff space X into E. The properties concern Ditkin's condition and bounded relative units. We show that these properties are shared by E and C(X, E). We also consider the relationship between these properties in the algebras E, B and $\~{B}$ that appear in the so-called admissible quadruples (X, E, B, $\~{B}$).

scholarly journals Duality properties of spaces of non-Archimedean valued functions

1987 ◽  
Vol 42 (1) ◽  
pp. 48-56
Author(s):  
W. Govaerts
Keyword(s):  
Uniform Convergence ◽  
Compact Hausdorff Space ◽  
Convex Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Complete Characterization ◽  
Bornological Space ◽  
Valued Field ◽  
Better Than

AbstractLet C(X, F) be the space of all continuous functions from the ultraregular compact Hausdorff space X into the separated locally K-convex space F; K is a complete, but not necessarily spherically complete, non-Archimedean valued field and C(X, F) is provided with the topology of uniform convergence on X We prove that C(X, F) is K-barrelled (respectively K-quasibarrelled) if and only if F is K-barrelled (respectively K-quasibarrelled) This is not true in the case of R or C-valued functions. No complete characterization of the K-bornological space C(X, F) is obtained, but our results are, nevertheless, slightly better than the Archimedean ones. Finally, we introduce a notion of K-ultrabornological spaces for K non-spherically complete and use it to study K-ultrabornological spaces C(X, F).

scholarly journals Fuzzy Ideal Supra Topological Spaces

2020 ◽  
Author(s):  
Fadhil Abbas
Keyword(s):  
Topological Space ◽  
Fuzzy Set ◽  
Basic Structure ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Local Function ◽  
Closed Sets ◽  
Fuzzy Ideal ◽  
Neighbourhood Structure

Abstract In this paper, we introduce the notion of fuzzy ideals in fuzzy supra topological spaces. The concept of a fuzzy s-local function is also introduced here by utilizing the s-neighbourhood structure for a fuzzy supra topological space. These concepts are discussed with a view to nd new fuzzy supra topologies from the original one. The basic structure, especially a basis for such generated fuzzy supra topologies and several relations between different fuzzy ideals and fuzzy supra topologies are also studied here. Moreover, we introduce a fuzzy set operator ΨS and study its properties. Finally, we introduce some sets of fuzzy ideal supra topological spaces (fuzzy *-supra dense-in-itself sets, fuzzy S*-supra closed sets, fuzzy *-supra perfect sets, fuzzy regular-I-supra closed sets, fuzzy-I-supra open sets, fuzzy semi-I-supra open sets, fuzzy pre-I-supra open sets, fuzzy α-I-supra open sets, fuzzy β-I-supra open sets) and study some characteristics of theses sets and then we introduce some fuzzy ideal supra continuous functions.

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