Compactoid and compact filters

Dolecki, Szymon; Greco, Gabriele H.; Lechicki, Alojzy

doi:10.2140/pjm.1985.117.69

Cited by 32 publications

(17 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…x ∈ lim G) iff N (x) ⊂ G. In a uniform space (X, U ), a filter G is called Cauchy iff for every U ∈ U there is G ∈ G with G × G ⊂ U . For a multifunction F and a filter G, the family F (G) : G ∈ G is a base of a filter, denoted by F G. Following definitions are from [4]. We say that a filter G is compactoid iff for every ultrafilter H ⊃ G, we have lim H = ∅.…”

Section: Subcontinuitymentioning

confidence: 99%

Subcontinuity

Holá

Novotný

2012

Mathematica Slovaca

View full text Add to dashboard Cite

ABSTRACT. We give interesting characterizations using subcontinuity. Let X, Y be topological spaces. We study subcontinuity of multifunctions from X to Y and its relations to local compactness, local total boundedness and upper semicontinuity. If Y is regular, then F is subcontinuous iff F is USCO. A uniform space Y is complete iff for every topological space X and for every net {F a }, F a ⊂ X × Y , of multifunctions subcontinuous at x ∈ X, uniformly convergent to F , F is subcontinuous at x. A Tychonoff space Y isČech-complete (resp. G m -space) iff for every topological space X and every multifunction F ⊂ X × Y the set of points of subcontinuity of F is a G δ -subset (resp. G m -subset) of X. There is a very natural extension of subcontinuity to multifunctions, introduced by Hrycay [22]1 . There is a large number of publications dealing with subcontinuity of either functions or multifunctions. Subcontinuity is often treated as a supplementary property, like before mentioned "upgrade" of closed graph property 2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 54C60, 54A25, 54E15. K e y w o r d s: subcontinuous, locally totally bounded, USCO multifunction, Vietoris topology, Hausdorff uniformity,Čech complete space. Both authors were supported by VEGA 2/0047/10. 1 This is the first reference we were able to find. Probably Smithson [34] did it independently in 1975, his definition is cited more frequently.

show abstract

Section: Subcontinuitymentioning

confidence: 99%

Subcontinuity

Holá

Novotný

2012

Mathematica Slovaca

View full text Add to dashboard Cite

show abstract

“…'Core-compact topologies have been called differently by different authors: hereditarily locally compactoid in [10], semi-locally bounded by J. R. Isbell [16], quasi locally compact by A. S. Ward [20], "condition C " by B. J. Day and G. M. Kelly [5].…”

Section: Fi9rmentioning

confidence: 99%

“…We shall first proceed to a review of some notions related to compactness (see [5,16,19,21] as well as [10] and its bibliographical comments).…”

Section: Notions Of Compactnessmentioning

confidence: 99%

“…For compactoid filters on regular topological spaces, we recall the following proposition [10]. It is very important to remember that compact families need not be filters; for example every family of compact sets is a compact family.…”

Section: Notions Of Compactnessmentioning

confidence: 99%

“…For 2Of course, the resulting Kuratowski and co-compact topologies are not Hausdorff, except for X = 0. 3Compact filters appear in [10], compact filters of open sets in [21] by P. Wilker and compact families of open sets in [5,16] under different names; compactoid filters may be found under a different name in [19] by F. Topsoe as well as in [ 10] and in some earlier papers of the authors.…”

Section: Notions Of Compactnessmentioning

confidence: 99%

See 2 more Smart Citations

When do the Upper Kuratowski Topology (Homeomorphically, Scott Topology) and the Co-compact Topology Coincide?

Dolecki¹,

Greco²,

Lechicki³

1995

Transactions of the American Mathematical Society

Self Cite

View full text Add to dashboard Cite

Abstract.A topology is called consonant if the corresponding upper Kuratowski topology on closed sets coincides with the co-compact topology, equivalently if each Scott open set is compactly generated. It is proved that Cechcomplete topologies are consonant and that consonance is not preserved by passage to G^-sets, quotient maps and finite products. However, in the class of the regular spaces, the product of a consonant topology and of a locally compact topology is consonant. The latter fact enables us to characterize the topologies generated by some T-convergences.

show abstract

Γ‐Inequalities and Stability of Generalized Extremal Convolutions

Dolecki

Guillerme

Lignola

et al. 1991

Mathematische Nachrichten

View full text Add to dashboard Cite

Convergence properties of convolutions, conjugations, compositions and polarities are expressed in terms of inequalities involving r-operators and generalized extremal convolutions. It is shown that r-inequalities amount to simple algebraic inequalities and (in some cases) to some min-max problems. Numerous applications are indicated, in particular, to continuity properties of the classical conjugation. IntroductionLooking over various reasonings of analysis, one is impressed by a number of arguments that amount to stability of certain connections between mathematical objects. The connections (the stability of which we are going to investigate) are expressed in terms of generalized extremal convolutions (GE-convolutions). They specialize to infimal and supremal convolutions and quasi-convolutions, to conjugations and quasi-conjugations, to compositions of relations and to polarities (Section 1). Consider non empty sets X I , . . . , X , and denote X : = fi X i and X(" := fi X , . i = l i = l , i + kConsider, as well, a function rp : R x R + R, where l? is the extended real line. We define the corresponding generalized infimal convolution (GI-convolution) as the following mapping I: R X x R X + R X :The resulting function is, in fact, an element of but we extend it, by constancy, to the whole of X (In what follows, we shall always tacitly extend considered functions to a common underlying space X). (GS)Symmetrically, the generalized supremal convolution (GS-convolution) is given by scf, 8) := SUP rpcf, g) * x k

show abstract

Compactoid and compact filters

Cited by 32 publications

References 8 publications

Subcontinuity

Subcontinuity

When do the Upper Kuratowski Topology (Homeomorphically, Scott Topology) and the Co-compact Topology Coincide?

Γ‐Inequalities and Stability of Generalized Extremal Convolutions

Contact Info

Product

Resources

About