Semiuniform Spaces Research Articles

Künneth theorems for Vietoris–Rips homology

We prove a Kunneth theorem for the Vietoris-Rips homology and cohomology of a semi-uniform space. We then interpret this result for graphs, where we show that the Kunneth theorem holds for graphs with respect to the strong graph product. We finish by computing the Vietoris-Rips cohomology of the torus endowed with diferent semi-uniform structures.

Rough semi-uniform spaces and its image proximities

<p style='text-indent:20px;'>In this paper, we introduce the concept of rough semi-uniform spaces as a supercategory of rough pseudometric spaces and approximation spaces. A completion of approximation spaces has been constructed using rough semi-uniform spaces. Applications of rough semi-uniform spaces in the construction of proximities of digital images is also discussed.</p>

Statistical convergence of sequences of functions with values in semi-uniform spaces

We study several kinds of statistical convergence of sequences of functions with values in semi-uniform spaces. Particularly, we generalize to statistical convergence the classical results of C. Arzela, Dini and P.S. Alexandroff, as well as their statistical versions studied in [Caserta A., Di Maio G., Kocinac L.D.R., {Statistical convergence in function spaces},. Abstr. Appl. Anal. 2011, Art. ID 420419, 11 pp.] and [Caserta A., Kocinac L.D.R., {On statistical exhaustiveness}, Appl. Math. Lett. 25 (2012), no. 10, 1447--1451].

Quasi-uniform Space

Abstract In this article, using mostly Pervin [9], Kunzi [6], [8], [7], Williams [11] and Bourbaki [3] works, we formalize in Mizar [2] the notions of quasiuniform space, semi-uniform space and locally uniform space. We define the topology induced by a quasi-uniform space. Finally we formalize from the sets of the form ((X \ Ω) × X) ∪ (X × Ω), the Csaszar-Pervin quasi-uniform space induced by a topological space.

Uniform Space

Abstract In this article, we formalize in Mizar [1] the notion of uniform space introduced by André Weil using the concepts of entourages [2]. We present some results between uniform space and pseudo metric space. We introduce the concepts of left-uniformity and right-uniformity of a topological group. Next, we define the concept of the partition topology. Following the Vlach’s works [11, 10], we define the semi-uniform space induced by a tolerance and the uniform space induced by an equivalence relation. Finally, using mostly Gehrke, Grigorieff and Pin [4] works, a Pervin uniform space defined from the sets of the form ((X\A) × (X\A)) ∪ (A×A) is presented.

Completion of quasi-topological groups

A topology of a quasi-topological group is induced by several natural semi-uniformities, namely right, left, two-sided and Roelcke semi-uniformities. A quasi-topological group is called complete if every Cauchy (in some sense—we examine several generalizations of Cauchy properties) filter on the two-sided semi-uniformity converges. We use the theory of Hausdorff complete semi-uniform spaces, see [B. Batíková, Completion of semi-uniform spaces, Appl. Categor. Struct. 15 (2007) 483–491], and show that Hausdorff complete quasi-topological groups form an epireflective subcategory of Hausdorff quasi-topological groups. But the reflection arrows need not be embeddings. For several types of Cauchy-like properties we show examples of quasi-topological groups that cannot be embedded into a complete group.

Fibrewise covering uniformities and completions

By using coverings we introduce the concepts of fibrewise covering uniform space and its generalizations (fibrewise generalized uniform space and fibrewise semi-uniform space), and study the fibrewise completions of fibrewise generalized uniform spaces and fibrewise semi-uniform spaces.

Completion of Semi-uniform Spaces

The category of all Hausdorff complete t-semi-uniform spaces is shown to be epireflective in the category of all Hausdorff t-semi-uniform spaces but the reflection arrows need not be embeddings since there is no nontrivial epireflective subcategory of the category of all Hausdorff t-semi-uniform spaces in which all reflection arrows are embeddings (t-semi-uniform spaces are those semi-uniform spaces inducing a topology). On the other hand for every t-semi-uniform space X there exist a minimal and a maximal completion containing X as a dense subspace. The second one is an almost reflection in complete spaces, i.e., every uniformly continuous mapping on X to a complete semi-uniform space can be extended (as a uniformly continuous map) onto the completion.

The Nonstandard Theory of Semi‐Uniform Spaces

Mathematical Logic QuarterlyVolume 24, Issue 15-16 p. 237-256 Article The Nonstandard Theory of Semi-Uniform Spaces Robert A. Herrmann, Robert A. Herrmann Annapolis, Maryland (U.S.A.)Search for more papers by this author Robert A. Herrmann, Robert A. Herrmann Annapolis, Maryland (U.S.A.)Search for more papers by this author First published: 1978 https://doi.org/10.1002/malq.19780241503AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Volume24, Issue15-161978Pages 237-256 RelatedInformation

Semiuniform Spaces and Topological Homeomorphism Groups

Semiuniform Spaces and Topological Homeomorphism Groups

Semiuniform spaces and topological homeomorphism groups

A well-known sufficient condition that a group of homeomorphisms, H H , from a topological space X X onto itself be a topological group relative to the topology of pointwise convergence is that X X be uniformizable and H H be equicontinuous. In this paper we prove an analogous condition in which the space X X is assumed to be only regular instead of completely regular (uniformizable). This is accomplished by means of the concepts of semiuniformity and semiequicontinuity introduced here.