The anti-Specker property, positivity, and total boundedness

2010 ◽  
Vol 56 (4) ◽  
pp. 434-441 ◽  
Author(s):  
Douglas Bridges ◽  
Hannes Diener
Keyword(s):  
Total Boundedness

scholarly journals Compactifications of totally bounded quasi-uniform spaces

1986 ◽  
Vol 28 (1) ◽  
pp. 31-36 ◽  
Author(s):  
P. Fletcher ◽  
W. F. Lindgren
Keyword(s):  
Hausdorff Space ◽  
Uniform Space ◽  
Continuous Extension ◽  
Sufficient Condition ◽  
Uniform Spaces ◽  
Completely Regular ◽  
Totally Bounded ◽  
Continuous Map ◽  
Total Boundedness ◽  
Uniformly Continuous

The notation and terminology of this paper coincide with that of reference [4], except that here the term, compactification, refers to a T1-space. It is known that a completely regular totally bounded Hausdorff quasi-uniform space (X, ) has a Hausdorff compactification if and only if contains a uniformity compatible with ℱ() [4, Theorem 3.47]. The use of regular filters by E. M. Alfsen and J. E. Fenstad [1] and O. Njåstad [5], suggests a construction of a compactification, which differs markedly from the construction obtained in [4]. We use this construction to show that a totally bounded T1 quasi-uniform space has a compactification if and only if it is point symmetric. While it is pleasant to have a characterization that obtains for all T1-spaces, the present construction has several further attributes. Unlike the compactification obtained in [4], the compactification given here preserves both total boundedness and uniform weight, and coincides with the uniform completion when the quasi-uniformity under consideration is a uniformity. Moreover, any quasi-uniformly continuous map from the underlying quasi-uniform space of the compactification onto any totally bounded compact T1-space has a quasi-uniformly continuous extension to the compactification. If is the Pervin quasi-uniformity of a T1-space X, the compactification we obtain is the Wallman compactification of (X, ℱ ()). It follows that our construction need not provide a Hausdorff compactification, even when such a compactification exists; but we obtain a sufficient condition in order that our compactification be a Hausdorff space and note that this condition is satisfied by all uniform spaces and all normal equinormal quasi-uniform spaces. Finally, we note that our construction is reminiscent of the completion obtained by Á. Császár for an arbitrary quasi-uniform space [2, Section 3]; in particular our Theorem 3.7 is comparable with the result of [2, Theorem 3.5].

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 Monotonicity and total boundedness in spaces of “measurable” functions

2017 ◽  
Vol 67 (6) ◽  
Author(s):  
Diana Caponetti ◽  
Alessandro Trombetta ◽  
Giulio Trombetta
Keyword(s):  
Total Boundedness ◽  
Measurable Functions

AbstractWe define and study the moduli

scholarly journals Duality and quasi-normability for complexity spaces

2002 ◽  
Vol 3 (1) ◽  
pp. 91 ◽  
Author(s):  
Salvador Romaguera ◽  
M.P. Schellekens
Keyword(s):  
Uniform Convergence ◽  
Large Class ◽  
Function Space ◽  
Metric Space ◽  
Normed Space ◽  
Complexity Analysis ◽  
Metric Properties ◽  
Total Boundedness ◽  
Semilinear Space ◽  
Lower Boundedness

<p>The complexity (quasi-metric) space was introduced in [23] to study complexity analysis of programs. Recently, it was introduced in [22] the dual complexity (quasi-metric) space, as a subspace of the function space [0,) <sup>ω</sup>. Several quasi-metric properties of the complexity space were obtained via the analysis of its dual.</p> <p>We here show that the structure of a quasi-normed semilinear space provides a suitable setting to carry out an analysis of the dual complexity space. We show that if (E,) is a biBanach space (i.e., a quasi-normed space whose induced quasi-metric is bicomplete), then the function space (B*<sub>E</sub>, <sub>B*</sub> ) is biBanach, where B*<sub>E</sub> = {f :   E  Σ<sup>∞</sup><sub>n=0</sub> 2<sup>-n</sup>( V ) }  and <sub>B*</sub> = Σ<sup>∞</sup><sub>n=0</sub> 2<sup>-n</sup> We deduce that the dual complexity space admits a structure of quasinormed semlinear space such that the induced quasi-metric space is order-convex, upper weightable and Smyth complete, not only in the case that this dual is a subspace of [0,)<sup>ω</sup> but also in the general case that it is a subspace of F<sup>ω</sup> where F is any biBanach normweightable space. We also prove that for a large class of dual complexity (sub)spaces, lower boundedness implies total boundedness. Finally, we investigate completeness of the quasi-metric of uniform convergence and of the Hausdorff quasi-pseudo-metric for the dual complexity space, in the context of function spaces and hyperspaces, respectively.</p>

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