Atoms, anti-atoms and complements in the lattice of quasi-uniformities

Jager, Eliza P. de; Künzi, Hans-Peter A.

doi:10.1016/j.topol.2004.10.021

Cited by 10 publications

(5 citation statements)

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“…Otherwise it will be called nonsymmetric. It is readily seen (compare with [4]) that the supremum and infimum of an arbitrary family of uniformities in (q(X), ⊆) is a uniformity.…”

Section: Preliminariesmentioning

confidence: 99%

“…(For instance, we can choose any proximally discrete anti-atom of (q(X), ⊆)-compare [4].) We are going to show that the quasiuniformities Q ω × Q and Q −1 × Q ω are both proximally symmetric, but that their supremum Q −1 × Q is not proximally symmetric.…”

Section: Examplementioning

confidence: 99%

“…A quasi-uniformity V is a complement of a quasi-uniformity U in the lattice (q(X), ⊆) provided that V ∨ U = D and V ∧ U = I. Let us mention that some results regarding complements in (q(X), ⊆) are contained in [4].…”

Section: More About Quasi-uniform Complementsmentioning

confidence: 99%

“…Let p ∈ ω such that p 2 and (X ∩ [0, 1 p−1 )) ⊆ U n (0). Then 0 ∈ U n ( 1 p ) and hence (X It is known that each compatible uniformity on a resolvable completely regular space X is complemented in (q(X), ⊆) (see [4,Proposition 8]). This suggests the following natural question.…”

Section: Corollarymentioning

confidence: 99%

“…In the following we continue our study [4,5] of the set q(X) of all quasi-uniformities on a set X, partially ordered under set-theoretic inclusion ⊆. It is well known that (q(X), ⊆) is a complete lattice [8, p. 2].…”

Section: Introductionmentioning

confidence: 99%

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Infima and complements in the lattice of quasi-uniformities

Jager

Künzi

2007

Topology and its Applications

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We show that the infimum of any family of proximally symmetric quasi-uniformities is proximally symmetric, while the supremum of two proximally symmetric quasi-uniformities need not be proximally symmetric. On the other hand, the supremum of any family of transitive quasi-uniformities is transitive, while there are transitive quasi-uniformities whose infimum with their conjugate quasi-uniformity is not transitive. Moreover we present two examples that show that neither the supremum topology nor the infimum topology of two transitive topologies need be transitive. Finally, we prove that several operations one can perform on and between quasi-uniformities preserve the property of having a complement.

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“…Otherwise it will be called nonsymmetric. It is readily seen (compare with [4]) that the supremum and infimum of an arbitrary family of uniformities in (q(X), ⊆) is a uniformity.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Examplementioning

confidence: 99%

Section: More About Quasi-uniform Complementsmentioning

confidence: 99%

Section: Corollarymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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