scholarly journals T3 AND T4-objects in the topological category of cauchy spaces

2017 ◽  
Vol 66 (1) ◽  
pp. 29-42
Author(s):  
KULA Muammer
Keyword(s):  
Topological Category ◽  
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 .

The Disc Embedding Theorem

2021 ◽  
Keyword(s):  
Graduate Students ◽  
Euclidean Space ◽  
Iterative Process ◽  
Embedding Theorem ◽  
Space Theory ◽  
Topological Category ◽  
Surgery Theory ◽  
Smooth Structures ◽  
Decomposition Space ◽  
Exotic Smooth Structures

The disc embedding theorem provides a detailed proof of the eponymous theorem in 4-manifold topology. The theorem, due to Michael Freedman, underpins virtually all of our understanding of 4-manifolds in the topological category. Most famously, this includes the 4-dimensional topological Poincaré conjecture. Combined with the concurrent work of Simon Donaldson, the theorem reveals a remarkable disparity between the topological and smooth categories for 4-manifolds. A thorough exposition of Freedman’s proof of the disc embedding theorem is given, with many new details. A self-contained account of decomposition space theory, a beautiful but outmoded branch of topology that produces non-differentiable homeomorphisms between manifolds, is provided. Techniques from decomposition space theory are used to show that an object produced by an infinite, iterative process, which we call a skyscraper, is homeomorphic to a thickened disc, relative to its boundary. A stand-alone interlude explains the disc embedding theorem’s key role in smoothing theory, the existence of exotic smooth structures on Euclidean space, and all known homeomorphism classifications of 4-manifolds via surgery theory and the s-cobordism theorem. The book is written to be accessible to graduate students working on 4-manifolds, as well as researchers in related areas. It contains over a hundred professionally rendered figures.

Homotopy Theory of Diagrams and CW-Complexes Over a Category

1991 ◽  
Vol 43 (4) ◽  
pp. 814-824 ◽  
Author(s):  
Robert J. Piacenza
Keyword(s):  
Classical Theory ◽  
Homotopy Theory ◽  
Homotopy Groups ◽  
Weak Equivalence ◽  
Equivariant Homotopy ◽  
Topological Category ◽  
Sheaf Theory ◽  
Closed Model ◽  
Base Category ◽  
Closed Model Category

The purpose of this paper is to introduce the notion of a CW complex over a topological category. The main theorem of this paper gives an equivalence between the homotopy theory of diagrams of spaces based on a topological category and the homotopy theory of CW complexes over the same base category.A brief description of the paper goes as follows: in Section 1 we introduce the homotopy category of diagrams of spaces based on a fixed topological category. In Section 2 homotopy groups for diagrams are defined. These are used to define the concept of weak equivalence and J-n equivalence that generalize the classical definition. In Section 3 we adapt the classical theory of CW complexes to develop a cellular theory for diagrams. In Section 4 we use sheaf theory to define a reasonable cohomology theory of diagrams and compare it to previously defined theories. In Section 5 we define a closed model category structure for the homotopy theory of diagrams. We show this Quillen type homotopy theory is equivalent to the homotopy theory of J-CW complexes. In Section 6 we apply our constructions and results to prove a useful result in equivariant homotopy theory originally proved by Elmendorf by a different method.

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.

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