Cauchy spaces

1970 ◽  
Vol 187 (3) ◽  
pp. 175-186 ◽  
Author(s):  
J. F. Ramaley ◽  
Oswald Wyler
Keyword(s):  
Cauchy Spaces

scholarly journals Order compatibility for Cauchy spaces and convergence spaces

1987 ◽  
Vol 10 (2) ◽  
pp. 209-216
Author(s):  
D. C. Kent ◽  
Reino Vainio
Keyword(s):  
Uniform Convergence ◽  
Weak Order ◽  
Convergence Spaces ◽  
Convergence Structure ◽  
Cauchy Structure ◽  
Cauchy Spaces

A Cauchy structure and a preorder on the same set are said to be compatible if both arise from the same quasi-uniform convergence structure onX. Howover, there are two natural ways to derive the former structures from the latter, leading to “strong” and “weak” notions of order compatibility for Cauchy spaces. These in turn lead to characterizations of strong and weak order compatibility for convergence spaces.

scholarly journals Products of Composition and Differentiation between the Fractional Cauchy Spaces and the Bloch-Type Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
R. A. Hibschweiler
Keyword(s):  
Unit Disc ◽  
Cauchy Transforms ◽  
Type Spaces ◽  
Cauchy Spaces

The operators D C Φ and C Φ D are defined by D C Φ f = f ∘ Φ ′ and C Φ D f = f ′ ∘ Φ where Φ is an analytic self-map of the unit disc and f is analytic in the disc. A characterization is provided for boundedness and compactness of the products of composition and differentiation from the spaces of fractional Cauchy transforms F α to the Bloch-type spaces B β , where α > 0 and β > 0 . In the case β < 2 , the operator D C Φ : F α ⟶ B β is compact ⇔ D C Φ : F α ⟶ B β is bounded ⇔ Φ ′ ∈ B β , Φ Φ ′ ∈ B β and Φ ∞ < 1 . For β < 1 , C Φ D : F α ⟶ B β is compact ⇔ C Φ D : F α ⟶ B β is bounded ⇔ Φ ∈ B β and Φ ∞ < 1 .

On the Regularity of the Kowalsky Completion

1984 ◽  
Vol 36 (1) ◽  
pp. 58-70 ◽  
Author(s):  
Eva Lowen-Colebunders
Keyword(s):  
Uniform Convergence ◽  
Essential Role ◽  
Convergence Spaces ◽  
The Past ◽  
Cauchy Space ◽  
Cauchy Spaces

Cauchy spaces were introduced by Kowalsky in 1954 [9]. In that paper a first completion method for these spaces was given. In 1968 Keller [5] has shown that the Cauchy space axioms characterize the collections of Cauchy filters of uniform convergence spaces in the sense of [1]. Moreover in the completion theory of uniform convergence spaces the associated Cauchy structures play an essential role [12]. This fact explains why in the past ten years in the theory of Cauchy spaces, much attention has been given to the study of completions.

scholarly journals The category of cauchy spaces is cartesian closed

1987 ◽  
Vol 27 (2) ◽  
pp. 105-112 ◽  
Author(s):  
H.L. Bentley ◽  
H. Herrlich ◽  
E. Lowen-Colebunders
Keyword(s):  
Cartesian Closed ◽  
Cauchy Spaces

scholarly journals p-topological Cauchy completions

1999 ◽  
Vol 22 (3) ◽  
pp. 497-509
Author(s):  
J. Wig ◽  
D. C. Kent
Keyword(s):  
Equivalence Class ◽  
Topological Property ◽  
Convergence Space ◽  
Extension Theorems ◽  
General Properties ◽  
Continuous Map ◽  
Cauchy Space ◽  
Topological Completion ◽  
Cauchy Spaces ◽  
Natural Way

The duality between “regular” and “topological” as convergence space properties extends in a natural way to the more general properties “p-regular” and “p-topological.” Since earlier papers have investigated regular,p-regular, and topological Cauchy completions, we hereby initiate a study ofp-topological Cauchy completions. Ap-topological Cauchy space has ap-topological completion if and only if it is “cushioned,” meaning that each equivalence class of nonconvergent Cauchy filters contains a smallest filter. For a Cauchy space allowing ap-topological completion, it is shown that a certain class of Reed completions preserve thep-topological property, including the Wyler and Kowalsky completions, which are, respectively, the finest and the coarsestp-topological completions. However, not allp-topological completions are Reed completions. Several extension theorems forp-topological completions are obtained. The most interesting of these states that any Cauchy-continuous map between Cauchy spaces allowingp-topological andp′-topological completions, respectively, can always be extended to aθ-continuous map between anyp-topological completion of the first space and anyp′-topological completion of the second.

Cauchy spaces

1970 ◽  
Vol 187 (3) ◽  
pp. 187-199 ◽  
Author(s):  
J. F. Ramaley ◽  
Oswald Wyler
Keyword(s):  
Cauchy Spaces

scholarly journals Ordered Cauchy spaces

1985 ◽  
Vol 8 (3) ◽  
pp. 483-496 ◽  
Author(s):  
D. C. Kent ◽  
R. Vainio
Keyword(s):  
Total Boundedness ◽  
Cauchy Space ◽  
Cauchy Spaces

This paper is concerned with the notion of “ordered Cauchy space” which is given a simple internal characterization in Section 2. It gives a discription of the category of ordered Cauchy spaces which have ordered completions, and a construction of the “fine completion functor” on this category. Sections 4 through 6 deals with certain classes of ordered Cauchy spaces which have ordered completions; examples are given which show that the fine completion does not preserve such properties as uniformizability, regularity, or total boundedness. From these results, it is evident that a further study of ordered Cauchy completions is needed.

scholarly journals Completions of non-T2filter spaces

2005 ◽  
Vol 2005 (24) ◽  
pp. 4019-4027
Author(s):  
Nandita Rath
Keyword(s):  
Cauchy Spaces

The well-known completions ofT2Cauchy spaces andT2filter spaces are extended to the completions of non-T2filter spaces, and a completion functor on the category of all filter spaces is described.

A note on Cauchy spaces

2011 ◽  
Vol 133 (1-2) ◽  
pp. 14-32 ◽  
Author(s):  
Muammer Kula
Keyword(s):  
Cauchy Spaces
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