Function Space Topologies for Connectivity and Semi-Connectivity Functions

1966 ◽  
Vol 9 (3) ◽  
pp. 349-352 ◽  
Author(s):  
Somashekhar Amrith Naimpally
Keyword(s):  
Uniform Convergence ◽  
Function Space ◽  
Uniform Space ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Subsequent Paper ◽  
Graph Topology ◽  
Function Space Topologies

Let X and Y be topological spaces. If Y is a uniform space then one of the most useful function space topologies for the class of continuous functions on X to Y (denoted by C) is the topology of uniform convergence. The reason for this usefulness is the fact that in this topology C is closed in YX (see Theorem 9, page 227 in [2]) and consequently, if Y is complete then C is complete. In this paper I shall show that a similar result is true for the function space of connectivity functions in the topology of uniform convergence and for the function space of semi-connectivity functions in the graph topology when X×Y is completely normal. In a subsequent paper the problem of connected functions will be discussed.

scholarly journals Set-open topologies on function spaces

2018 ◽  
Vol 19 (1) ◽  
pp. 55
Author(s):  
Wafa Khalaf Alqurashi ◽  
Liaqat Ali Khan ◽  
Alexander V. Osipov
Keyword(s):  
Topological Space ◽  
Uniform Convergence ◽  
Uniform Space ◽  
Continuous Functions ◽  
Function Spaces ◽  
Topological Spaces ◽  
General Topological Space

<p>Let X and Y be topological spaces, F(X,Y) the set of all functions from X into Y and C(X,Y) the set of all continuous functions in F(X,Y). We study various set-open topologies tλ (λ ⊆ P(X)) on F(X,Y) and consider their existence, comparison and coincidence in the setting of Y a general topological space as well as for Y = R. Further, we consider the parallel notion of quasi-uniform convergence topologies Uλ (λ ⊆ P(X)) on F(X,Y) to discuss Uλ-closedness and right Uλ-K-completeness properties of a certain subspace of F(X,Y) in the case of Y a locally symmetric quasi-uniform space. We include some counter-examples to justify our comments.</p>

Uniformities on a Product

1972 ◽  
Vol 24 (3) ◽  
pp. 379-389 ◽  
Author(s):  
Anthony W. Hager
Keyword(s):  
Topological Space ◽  
Cardinal Number ◽  
Uniform Space ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Infinite Cardinal ◽  
The Real ◽  
Completely Regular

All topological spaces shall be uniformizable (completely regular Hausdorff). A uniformity on X shall be viewed as a collection μ of coverings of X, via the manner of Tukey [20] and Isbell [16], and the associated uniform space denoted μX. Given the uniformizable topological space X, we shall be concerned with compatible uniformities as follows (discussed more carefully in § 1). The fine uniformity α (finest compatible with the topology); the “cardinal reflections“ αm of α (m an infinite cardinal number) ; αc, the weak uniformity generated by the real-valued continuous functions.With μ standing, generically, for one of these uniformities, we consider the question: when is μ(X × Y) = μX × μY For μ = αℵ0 (the finest compatible precompact uniformity), the problem is equivalent to that of whenβ(X × Y) = βX × βY,β denoting Stone-Cech compactification; this is answered by the theorem of Glicksberg [9]. For μ = α, we have Isbell's generalization [16, VI1.32].

Local Compactness in Set Valued Function Spaces

1976 ◽  
Vol 19 (2) ◽  
pp. 193-198 ◽  
Author(s):  
Saroop K. Kaul
Keyword(s):  
Uniform Convergence ◽  
Uniform Space ◽  
Continuous Functions ◽  
Function Spaces ◽  
Range Space ◽  
Locally Compact ◽  
Local Compactness

Recently Hunsaker and Naimpally [2] have proved: The pointwise closure of an equicontinuous family of point compact relations from a compact T2-space to a locally compact uniform space is locally compact in the topology of uniform convergence. This is a generalization of the same result of Fuller [1] for single valued continuous functions.For a range space which is locally compact normal and uniform theorem B below is an improvement on the result of Hunsaker and Naimpally quoted above [see Remark 3 at the end of this paper].

scholarly journals Quasi-uniform convergence topologies on function spaces- Revisited

2017 ◽  
Vol 18 (2) ◽  
pp. 301
Author(s):  
Wafa Khalaf Alqurash ◽  
Liaqat Ali Khan
Keyword(s):  
Topological Space ◽  
Uniform Convergence ◽  
Uniform Space ◽  
Continuous Functions ◽  
Function Spaces

Let X and Y be topological space and F(X,Y) the set of all functions from X into Y. We study various quasi-uniform convergence topologies U_{A} (A⊆P(X)) on F(X,Y) and their comparison in the setting of Y a quasi-uniform space. Further, we study U_{A}-closedness and right K-completeness properties of certain subspaces of generalized continuous functions in F(X,Y) in the case of Y a locally symmetric quasi-uniform space or a locally uniform space.

scholarly journals A study of topological structures on equi-continuous mappings

2021 ◽  
Vol 40 (2) ◽  
pp. 335-354
Author(s):  
Ankit Gupta ◽  
Ratna Dev Sarma
Keyword(s):  
Topological Space ◽  
Function Space ◽  
Uniform Space ◽  
Function Spaces ◽  
Topological Structures ◽  
Continuous Mappings ◽  
Function Space Topologies ◽  
Admissibility Of Function Spaces

Function space topologies are developed for EC(Y,Z), the class of equi-continuous mappings from a topological space Y to a uniform space Z. Properties such as splittingness, admissibility etc. are defined for such spaces. The net theoretic investigations are carried out to provide characterizations of splittingness and admissibility of function spaces on EC(Y,Z). The open-entourage topology and pointtransitive-entourage topology are shown to be admissible and splitting respectively. Dual topologies are defined. A topology on EC(Y,Z) is found to be admissible (resp. splitting) if and only if its dual is so.

scholarly journals An Ascoli theorem for sequential spaces

2001 ◽  
Vol 26 (5) ◽  
pp. 303-315 ◽  
Author(s):  
Gert Sonck
Keyword(s):  
Function Space ◽  
Uniform Space ◽  
Continuous Functions ◽  
Space Structure ◽  
Sequential Space ◽  
Continuous Convergence ◽  
Natural Function ◽  
Sequential Spaces

Ascoli theorems characterize “precompact” subsets of the set of morphisms between two objects of a category in terms of “equicontinuity” and “pointwise precompactness,” with appropriate definitions of precompactness and equicontinuity in the studied category. An Ascoli theorem is presented for sets of continuous functions from a sequential space to a uniform space. In our development we make extensive use of the natural function space structure for sequential spaces induced by continuous convergence and define appropriate concepts of equicontinuity for sequential spaces. We apply our theorem in the context ofC*-algebras.

scholarly journals R-continuous functions

1992 ◽  
Vol 15 (1) ◽  
pp. 57-64 ◽  
Author(s):  
Ch. Konstadilaki-Savvapoulou ◽  
D. Janković
Keyword(s):  
Continuous Functions ◽  
Strong Form ◽  
Topological Spaces ◽  
Strong Continuity ◽  
Closed Graph ◽  
Continuity Properties

A strong form of continuity of functions between topological spaces is introduced and studied. It is shown that in many known results, especially closed graph theorems, functions under consideration areR-continuous. Several results in the literature concerning strong continuity properties are generalized and/or improved.

scholarly journals On Softβ-Open Sets and Softβ-Continuous Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Metin Akdag ◽  
Alkan Ozkan
Keyword(s):  
Continuous Functions ◽  
Soft Set ◽  
Topological Spaces

We introduce the concepts softβ-interior and softβ-closure of a soft set in soft topological spaces. We also study softβ-continuous functions and discuss their relations with soft continuous and other weaker forms of soft continuous functions.

scholarly journals A note on weakly θ-continuous functions

1989 ◽  
Vol 12 (1) ◽  
pp. 9-13 ◽  
Author(s):  
M. Mrševic ◽  
I. L. Reilly
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
Topological Spaces ◽  
New Class ◽  
Weakly Continuous ◽  
Class Of Functions

Recently a new class of functions between topological spaces, called weaklyθ-continuous functions, has been introduced and studied. In this paper we show how an appropriate change of topology on the domain of a weaklyθ-continuous function reduces it to a weakly continuous function. This paper examines some of the consequences of this result.

scholarly journals Adelic descent theory

2017 ◽  
Vol 153 (8) ◽  
pp. 1706-1746
Author(s):  
Michael Groechenig
Keyword(s):  
Vector Bundles ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Reductive Algebraic Group ◽  
Perfect Complexes ◽  
Descent Theory ◽  
Closed Fields ◽  
Ring Of Continuous Functions ◽  
Reconstruction Theorem ◽  
Algebraically Closed Fields

A result of André Weil allows one to describe rank $n$ vector bundles on a smooth complete algebraic curve up to isomorphism via a double quotient of the set $\text{GL}_{n}(\mathbb{A})$ of regular matrices over the ring of adèles (over algebraically closed fields, this result is also known to extend to $G$-torsors for a reductive algebraic group $G$). In the present paper we develop analogous adelic descriptions for vector and principal bundles on arbitrary Noetherian schemes, by proving an adelic descent theorem for perfect complexes. We show that for Beilinson’s co-simplicial ring of adèles $\mathbb{A}_{X}^{\bullet }$, we have an equivalence $\mathsf{Perf}(X)\simeq |\mathsf{Perf}(\mathbb{A}_{X}^{\bullet })|$ between perfect complexes on $X$ and cartesian perfect complexes for $\mathbb{A}_{X}^{\bullet }$. Using the Tannakian formalism for symmetric monoidal $\infty$-categories, we conclude that a Noetherian scheme can be reconstructed from the co-simplicial ring of adèles. We view this statement as a scheme-theoretic analogue of Gelfand–Naimark’s reconstruction theorem for locally compact topological spaces from their ring of continuous functions. Several results for categories of perfect complexes over (a strong form of) flasque sheaves of algebras are established, which might be of independent interest.

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