On semi-uniformities

Steiner, A. K.; Steiner, E. F.

doi:10.4064/fm-83-1-47-58

Cited by 16 publications

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“…MORITA assumes that the basic set is topological and considers c~llect~ions of open covers for this generalization. STEINER and STEINER rediscover MORITA'S regular T-uniformities in their paper [13]. Even though the STEINER'S basic struct,ure, the semi-uniform space, is not a t first considered to be topological in nature, they apply this structure to obtain a compatible regular topology and show that their semi-uniformity is a regular T-uniformity of MORITA.…”

Section: Introductionmentioning

confidence: 99%

“…Not, a great deal appears to be known about semi-uniform spaces since only the papers [ll], [13] and an important coiit,ribution to extension theory by WOOTEN [19] are known to t,he present, author. One of t'he goals of t,his paper is to strengt'hen the STEINER'S claim and to further investjigate t,he analogies between semi-uniform and uniform structures.…”

Section: Introductionmentioning

confidence: 99%

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The Nonstandard Theory of Semi‐Uniform Spaces

Herrmann

1978

Mathematical Logic Qtrly

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F E 9). It is well known that N u c S + 0 iff 9 has the finite intersection property.Further, we assume that all filter bases are non-trivial.Let 6 be a family of covers of the set X. If %, qL1 are elements of G and % refines then we denote this by 4Y < 4?ll. Also for A t X , let St(A, 42) = lJ ( U I [ U E %] A A [U n A + 011. Care should be taken not to confuse this "star" operator with the standard part operator of LUXEMBURG [9] and others. D e f i n i t i o n 2.1. A family of covers 6 of X is a semi-uniform structure if (1) for each ' ?/ E G there exists 42' E G such that for each U' E W there exists some(2) %, @' E G, then there exists some W' E 6 such that W' refines both %Y and %;(3) 42 E E and % refines @', then W E 6. Clearly, if G is a semi-uniform structure, then 6 is a filter with respect to refinement. We say that a set X together with a semi-uniformity G on X is a semi-uniform space.The following result is obtained directly from the definitions and is useful when we compare MORITA'S definitions and results with those of STEINER and STEINER.TJ E qL and a'' E G such that St(lJ', $2") c U ;T h e o r e m 2.1. Let G be a semi-uniformity on X .(i) If B c G i s a base for G, thetb 23 has propertzes (1) and (2) of defitiition 2.1.(ii) If % has properties (1) and (2) of defi?iitio?i 2 1, then (8) i s a semi-uniform stntcture 018 X with base 93. Cauchy familiesI n [13], a set F c 9 ( X ) is called a Cauchy family (with respect to a semi-uniformity G = (93)) if F has t,he finite int,ersection property and for each @ E G t'here exist,s some F E F and a1 E G such that St,(F, el) c U for some U E @. It is easily shown that tJhis definit,ion is equivalent to MORITA'S [ l l ] which states that 9 is a Cauchy family if F has the finite intersection propert,y and for each q/ E B there exist, F E F and E B such that St(F, qL1) c U for some U CI a. Two Cauchy families 3, 3 are equivalent (writ,t,en 9 -9) if for each F E F and @ E G = (B) there esist, G E 9 and %l E G such t,hat St(G, qL1) c St, (F, 92). Again bhis definition is equivalent to MORITA'S which states that 3 -9 if for each F E S and E B there exist G E 9 and E 23 such that St(G, q/J c St, (F, ?/). MORITA shows that the equivalence of Cauchy families is an equivalence relation. Throughout the remainder of t,his paper, unless ot)herwise mentioned, we shall assume that a semi-uniform space, ( X , B), is a set' X wit'h a semi-uniform base B. Hence (X, (123)) is a semi-uniform space in t,he

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Nonstandard Theory of Semi‐Uniform Spaces

Herrmann

1978

Mathematical Logic Qtrly

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“…If X is a space and C is a semi-uniformity on X, then (X,C) is said to be a semi-uniform space. The following can be found in [29]. coverings.…”

Section: B Semi-uniform Structuresmentioning

confidence: 99%

“…These spaces are called semi-uniform spaces [29]. They are, in particular, generalizations of uniform spaces.…”

Section: Preliminariesmentioning

confidence: 99%

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On the completion of topological structures and the extension of associated mappings

Wooten

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This dissertation was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted.The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1. The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity.2. When an image on the film is obliterated with a large round black mark, it is an indication that the photographer suspected that the copy may have moved during exposure and thus cause a blurred image. You will find a good image of the page in the adjacent frame.3. When a map, drawing or chart, etc., was part of the material being photographed the photographer followed a definite method in "sectioning" the material. It is customary to begin photoing at the upper left hand corner of a large sheet and to continue photoing from left to right in equal sections with a small overlap. If necessary, sectioning is continued again -beginning below the first row and continuing on until complete.4. The majority of users indicate that the textual content is of greatest value, however, a somewhat higher quality reproduction could be made from "photographs" if essential to the understanding of the dissertation. Silver prints of "photographs" may be ordered at additional charge by writing the Order Department, giving the catalog number, title, author and specific pages you wish reproduced.

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Extensions of Maps from Dense Subspaces

Bentley

2001

Categorical Perspectives

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On semi-uniformities

Cited by 16 publications

References 0 publications

The Nonstandard Theory of Semi‐Uniform Spaces

The Nonstandard Theory of Semi‐Uniform Spaces

On the completion of topological structures and the extension of associated mappings

Extensions of Maps from Dense Subspaces

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