Regular completions of uniform convergence spaces

1974 ◽  
Vol 11 (3) ◽  
pp. 413-424 ◽  
Author(s):  
R.J. Gazik ◽  
D.C. Kent ◽  
G.D. Richardson
Keyword(s):  
Uniform Convergence ◽  
Real Line ◽  
Dual Space ◽  
Function Algebra ◽  
Continuous Functions ◽  
Convergence Space ◽  
Uniform Spaces ◽  
Convergence Spaces ◽  
Uniform Convergence Space ◽  
Uniformly Continuous

A regular completion with universal property is obtained for each member of the class of u–embedded uniform convergence spaces, a class which includes the Hausdorff uniform spaces. This completion is obtained by embedding each u-embedded uniform convergence space (X, I) into the dual space of a complete function algebra composed of the uniformly continuous functions from (X, I) into the real line.

scholarly journals Remarks on uniformly symmetrically continuous functions

2016 ◽  
Vol 09 (03) ◽  
pp. 1650069
Author(s):  
Tammatada Khemaratchatakumthorn ◽  
Prapanpong Pongsriiam
Keyword(s):  
Real Line ◽  
Nonempty Subset ◽  
Continuous Functions ◽  
The Real ◽  
Definition Of ◽  
Symmetrically Continuous Functions ◽  
Uniformly Continuous

We give the definition of uniform symmetric continuity for functions defined on a nonempty subset of the real line. Then we investigate the properties of uniformly symmetrically continuous functions and compare them with those of symmetrically continuous functions and uniformly continuous functions. We obtain some characterizations of uniformly symmetrically continuous functions. Several examples are also given.

scholarly journals On a Durrmeyer-type modification of the Exponential sampling series

2020 ◽  
Author(s):  
Carlo Bardaro ◽  
Ilaria Mantellini
Keyword(s):  
Asymptotic Formula ◽  
Uniform Convergence ◽  
Modulus Of Continuity ◽  
Continuous Functions ◽  
Convergence Properties ◽  
Sampling Series ◽  
Quantitative Results ◽  
Uniformly Continuous

Abstract In this paper we introduce the exponential sampling Durrmeyer series. We discuss pointwise and uniform convergence properties and an asymptotic formula of Voronovskaja type. Quantitative results are given, using the usual modulus of continuity for uniformly continuous functions. Some examples are also described.

scholarly journals ON HOMOLOGICAL PROPERTIES FOR SOME MODULES OF UNIFORMLY CONTINUOUS FUNCTIONS OVER CONVOLUTION ALGEBRAS

2011 ◽  
Vol 84 (2) ◽  
pp. 177-185
Author(s):  
RASOUL NASR-ISFAHANI ◽  
SIMA SOLTANI RENANI
Keyword(s):  
Compact Group ◽  
Group Algebra ◽  
Dual Space ◽  
Continuous Functions ◽  
Locally Compact Group ◽  
Measure Algebra ◽  
Locally Compact ◽  
Convolution Algebras ◽  
Uniformly Continuous

AbstractFor a locally compact group G, let LUC(G) denote the space of all left uniformly continuous functions on G. Here, we investigate projectivity, injectivity and flatness of LUC(G) and its dual space LUC(G)* as Banach left modules over the group algebra as well as the measure algebra of G.

scholarly journals The order topology for function lattices and realcompactness

1981 ◽  
Vol 4 (2) ◽  
pp. 289-304 ◽  
Author(s):  
W. A. Feldman ◽  
J. F. Porter
Keyword(s):  
Uniform Convergence ◽  
Continuous Functions ◽  
Order Topology ◽  
Convergence Space ◽  
Vector Lattices ◽  
Convergence Structure ◽  
Relative Uniform Convergence ◽  
Compact Convergence

A latticeK(X,Y)of continuous functions on spaceXis associated to each compactificationYofX. It is shown forK(X,Y)that the order topology is the topology of compact convergence onXif and only ifXis realcompact inY. This result is used to provide a representation of a class of vector lattices with the order topology as lattices of continuous functions with the topology of compact convergence. This class includes everyC(X)and all countably universally complete function lattices with 1. It is shown that a choice ofK(X,Y)endowed with a natural convergence structure serves as the convergence space completion ofVwith the relative uniform convergence.

Equicontinuity and -recursive transformation groups

1965 ◽  
Vol 61 (2) ◽  
pp. 333-336 ◽  
Author(s):  
Janet Allsbrook ◽  
R. W. Bagley
Keyword(s):  
Phase Space ◽  
Transformation Group ◽  
Transformation Groups ◽  
Almost Periodic ◽  
Convergence Space ◽  
Topological Transformation ◽  
Uniform Convergence Space ◽  
Left And Right ◽  
Uniformly Continuous ◽  
Special Case

In this paper we obtain results on equicontinuity and apply them to certain recursive properties of topological transformation groups (X, T, π) with uniform phase space X. For example, in the special case that each transition πt is uniformly continuous we consider the transformation group (X,Ψ,ρ), where Ψ is the closure of {πt|t∈T} in the space of all unimorphisms of X onto X with the topology of uniform convergence (space index topology) and p(x, φ) = φ(x) for (x, φ)∈ X × Ψ. (See (1), page 94, 11·18.) If π is a mapping on X × T we usually write ‘xt’ for ‘π(x, t)’ and ‘AT’ for ‘π(A × T)’ where A ⊂ X. In this case we obtain the following results:I. If (X, T, π) is almost periodic [regularly almost periodic] and πxis equicontinuous, then (X,Φ,ρ) is almost periodic [regularly almost periodic]II. Let A be a compact subset of X such that. If the left and right uniformities of T are equal and (X, T, π) is almost periodic [regularly almost periodic], then (X, T, π) is almost periodic [regularly almost periodic].

A non-constant invariant function for certain ergodic flows

2008 ◽  
Vol 28 (3) ◽  
pp. 1031-1035
Author(s):  
SOL SCHWARTZMAN
Keyword(s):  
Vector Space ◽  
Real Line ◽  
Differentiable Manifold ◽  
Invariant Function ◽  
Continuous Functions ◽  
The Real ◽  
Almost All ◽  
Uniformly Continuous

AbstractLet U be the vector space of uniformly continuous real-valued functions on the real line $\mathbb {R}$ and let U0 denote the subspace of U consisting of all bounded uniformly continuous functions. If X is a compact differentiable manifold and we are given a flow on X, then we associate with the flow a function F:X→H1(X,U/U0) that is invariant under the flow. We give examples for which the flow on X is ergodic but there is no λ∈H1(X,U/U0) such that F(p)=λ for almost all points p.

Hausdorff Distance and a Compactness Criterion for Continuous Functions

1986 ◽  
Vol 29 (4) ◽  
pp. 463-468 ◽  
Author(s):  
Gerald Beer
Keyword(s):  
Uniform Convergence ◽  
Hausdorff Distance ◽  
Metric Spaces ◽  
Continuous Functions ◽  
Compactness Criterion ◽  
Uniformly Continuous

AbstractLet 〈X, dx〉 and 〈Y, dY〉 be metric spaces and let hp denote Hausdorff distance in X x Y induced by the metric p on X x Y given by p[(x1, y1), (x2, y2)] = max ﹛dx(x1, x2),dY(y1, y2)﹜- Using the fact that hp when restricted to the uniformly continuous functions from X to Y induces the topology of uniform convergence, we exhibit a natural compactness criterion for C(X, Y) when X is compact and Y is complete.

scholarly journals On the Topological and Uniform Structure of Diversities

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Andrew Poelstra
Keyword(s):  
Uniform Convergence ◽  
Natural Transformation ◽  
Uniform Continuity ◽  
Uniform Structure ◽  
Natural Analogue ◽  
Uniform Spaces ◽  
Tight Span ◽  
Cauchy Sequences ◽  
Analytical Properties ◽  
Uniformly Continuous

Diversities have recently been developed as multiway metrics admitting clear and useful notions of hyperconvexity and tight span. In this note, we consider the analytical properties of diversities, in particular the generalizations of uniform continuity, uniform convergence, Cauchy sequences, and completeness to diversities. We develop conformities, a diversity analogue of uniform spaces, which abstract these concepts in the metric case. We show that much of the theory of uniform spaces admits a natural analogue in this new structure; for example, conformities can be defined either axiomatically or in terms of uniformly continuous pseudodiversities. Just as diversities can be restricted to metrics, conformities can be restricted to uniformities. We find that these two notions of restriction, which are functors in the appropriate categories, are related by a natural transformation.

scholarly journals ON A CONJECTURE REGARDING THE UPPER GRAPH BOX DIMENSION OF BOUNDED SUBSETS OF THE REAL LINE

Fractals ◽  
2013 ◽  
Vol 21 (03n04) ◽  
pp. 1350017
Author(s):  
GIORGOS KELGIANNIS ◽  
VAIOS LASCHOS
Keyword(s):  
Real Line ◽  
Continuous Functions ◽  
Box Dimension ◽  
The Real ◽  
Bounded Set ◽  
Isolated Points ◽  
Uniformly Continuous

Let X ⊂ ℝ be a bounded set; we introduce a formula that calculates the upper graph box dimension of X (i.e. the supremum of the upper box dimension of the graph over all uniformly continuous functions defined on X). We demonstrate the strength of the formula by proving various corollaries. We conclude by constructing a collection of sets X with infinitely many isolated points, having upper box dimension a taking values from zero to one while their graph box dimension takes any value in [ max {2a, 1}, a + 1], answering this way, negatively to a conjecture posed.1

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