Hausdorff Uniform Space Research Articles

A development of contraction mapping principles on Hausdorff uniform spaces

Certain generalized Banach’s contraction mapping principles on metric spaces are unified and/or extended to Hausdorff uniform spaces. Also given are some relationships between the set of all cluster points of the Picard iterates and the set of all fixed points for the mapping. These are obtained by assuming that the latter set is nonempty and by considering certain “quasi"-contractive conditions. The ("quasi") contractive conditions are defined by using a suitable family of pseudometrics on the uniform space.

Regular completions of uniform convergence spaces

A regular completion with universal property is obtained for each member of the class ofu–embedded uniform convergence spaces, a class which includes the Hausdorff uniform spaces. This completion is obtained by embedding eachu-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.

An approach to fixed-point theorems on uniform spaces

Diaz and Metcalf [2] have some interesting results on the set of successive approximations of a self mapping which is either a nonexpansion or a contraction on a metric space with respect to the set of fixed points of the mapping. We have extended most of these results to a Hausdorff uniform space. We have also proved a Banach’s contraction mapping principle on a complete Hausdorff uniform space and indicated some applications in locally convex linear topological spaces.

On -compactlike spaces and reflective subcategories

We introduce the new topology on a topological space generated by the ▪-sets. For an extensive subcategory ▪ of a category A of Hausdorff spaces and continuous maps, we consider the subcategory ▪ of A determined by those members of A which are ▪-closed in their ▪-reflection spaces. It is shown that ▪ is also an extensive subcategory of A for every infinite cardinal number ▪ if A is hereditary. It is also shown that a completely regular space is ▪-closed in its Stone-Čech compactification iff it is ▪-compact, that a zero-dimensional space is ▪-closed in its maximal zero-dimensional compactification iff every maximal open closed filter with the ▪-intersection property is fixed, that a Hausdorff uniform space is ▪-closed in its completion iff every Cauchy filter with the ▪-intersection property is convergent, and that a Hausdorff space is ▪-closed in its Katětov extension iff every maximal open filter with the ▪-intersection property is convergent.

A fixed point theorem for a class of mappings

The Banach contraction mapping theorem states that every k-contractive mapping (0 < k < 1) of a non-empty complete metric space into itself has a unique fixed point. This result does not apply to R-contractive mappings considered in [3] and mappings on a complete Hausdorff topological group. We obtain a fixed point theorem for a class ~//(X) of functions on a complete Hausdorff uniform space (X, q/). Suppose now that q/is generated by a metric d for X. Then (a) °//(X) contains all of the k-contractive mappings on (X, d); (b) i fd is metrically convex, then ~//(X) contains all of the q~-contractive mappings on (X, d); (c) if (X, d) is compact, then q/(X) contains all of the contractive mappings on (X, d) [4]. For the terminology of uniform spaces, we refer the reader to [2] and [7]. Let (X, ay) be a non-empty uniform space. We shall use A to denote the diagonal of X x X. Let f be a function of X into itself. fn will denote the function on X defined by

Compactification and completion as absolute closure

It is well known that a Tychonoff (respectively, Hausdorff uniform) space is compact (resp., complete) iff it is absolutely closed, i.e., dense in no other such space; we shall sketch a proof of this to our purpose, which is: Given X X , we apply Zorn’s Lemma to obtain a space maximal with respect to the property of containing X X densely, thus compact or complete. For uniform spaces, the “maximal extension” is automatically the completion; for Tychonoff spaces, we must, and do, explicitly arrange things so that the “maximal extension” has the universal mapping property describing the Stone-Čech compactification. A variant of the construction yields the Hewitt realcompactification. A crucial point in the proofs is (of course) the exhibition of an upper bound for a chain; this is, in essence, a direct limit construction.

Condition for a function space to be locally compact

Let F be an equicontinuous family of functions from a compact Hausdorff space to a locally compact Hausdorff uniform space. In this paper we prove that the pointwise closure of F is locally compact relative to the topology of uniform con-