Construction of quasi-uniformities

Hunsaker, W.; Lindgren, William F.

doi:10.1007/bf01435413

Cited by 23 publications

(7 citation statements)

References 5 publications

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“…Thus the "natural" quasi-uniformity recently described by Nielsen and Sloyer [6] is simply the Csâszâr quasi-uniformity, hence is precisely the Pervin quasi-uniformity. This latter fact was noted by Hunsaker and Lindgren [4].…”

Section: (G W \G N+1 ) N ((Pi $ J\(u 5l N )) ^supporting

confidence: 54%

Uniqueness of Compatible Quasi-Uniformities

Votaw

1972

Can. math. bull.

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It is shown that a topological space has a unique compatible quasi-uniformity if its topology is finite. Examples are given to show the converse is false for T1 and for normal second countable spaces. Two sufficient conditions are given for a topological space to have a compatible quasi-uniformity strictly finer than the associated Császár-Pervin quasi-uniformity. These conditions are used to show that a Hausdorff, semi-regular or first countable T1 space has a unique compatible quasi-uniformity if and only if its topology is finite. Császár and Pervin described, in quite different ways, quasi-uniformities which induce a given topology. It is shown that, for a given topological space, Császár and Pervin described the same quasi-uniformity.

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Section: (G W \G N+1 ) N ((Pi $ J\(u 5l N )) ^supporting

confidence: 54%

Uniqueness of Compatible Quasi-Uniformities

Votaw

1972

Can. math. bull.

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“…Clearly, 6 i s a proximity on X i f f Q i s the unique t o t a l l y bounded uniformity which i s compatible with 6 ( [ 2 ] ) . Therefore, since the Pervln quasi-uniformity P is t o t a l l y bounded ( [ 6 ] , p .…”

Section: B$[x-a)mentioning

confidence: 99%

“…If 6 i s induced by Q , then Q i s said to be compatible with 6 . I t has been shown in [2] that for every quasi-proximity space (X, 6) there exists a unique t o t a l l y bounded quasi-uniformity Q, which i s t compatible with 6 . Moreover, Q. is the coarsest quasi-uniformity which is compatible with 6 and a base for Q, is the collection of all the n n sets of the form U B.M.…”

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confidence: 99%

Quasi proximal continuity

Lambrinos

1973

Bull. Austral. Math. Soc.

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Conditions are given, under which a quasi-proximally continuous function is quasi-uniformly continuous, or a continuous function is quasi-proximally continuous. Thus, basic results on uniform and proximal continuity are extended and some new results are obtained. Three results in the literature are shown to be false.According to [SI and [9], a quaei-proximity space is a pair (X, 6) , where X is a non empty set and 6 is a binary relation on the power set of X which satisfies: The pair (X, 6) becomes a proximity space when:Pervln showed in LSI that if the closure operator c is defined by c(A) = \x : {x}6A) , then each quasi-proximity space (X, 6) gives rise to a topology r(6) on X and that every topological space {X, r) is quasi-proximizable, that i s , there exists a quasi-proximity 6 on A T such that r(6) = r .Given a quasi-uniformity Q on X (for definitions see [6], [7] or [3]) the quasi-proximity 6 induced by Q is defined by: A6B iff

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“…In the sequel, we will make frequent use of the following theorems found in [4] and [5] respectively.…”

Section: Corollary If(xt)mentioning

confidence: 99%