Intrinsic approach spaces on domains

Colebunders, Eva; Wachter, Sarah De; Lowen, Bob

doi:10.1016/j.topol.2011.01.025

Cited by 10 publications

(10 citation statements)

References 9 publications

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“…From now on, we will simplify the terminology by omitting the words 'extended' and 'pseudo', so in this respect our terminology differs from common usage, but it does conform with the terminology used in Berckmoes et al (2011) and Colebunders et al (2011), and agrees with the practice in more categorically oriented papers on the subject such as Gutierres and Hofmann (2012). An extended pre-quasi-pseudo metric on a set X is usually a function q : X × X → [0, ∞] that vanishes on the diagonal: if q also satisfies the triangular inequality, it is called an extended quasi-pseudo metric, and if q satisfies both the triangular inequality and symmetry, it is called an extended pseudo metric.…”

Section: Introductionmentioning

confidence: 90%

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Fixed points of contractive maps on dcpo's

Colebunders

Wachter

Löwen

2013

Math. Struct. Comp. Sci.

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In this paper we study approach structures on dcpo's. A dcpo (X, 6) will be endowed with several other structures: the Scott topology; an approach structure generated by a collection of weightable quasi metrics on X; and a collection W of weights corresponding to the quasi metrics. Understanding the interaction between these structures on X will eventually lead to some fixed-point theorems for the morphisms in the category of approach spaces, which are called contractions. Existing fixed-point theorems on both monotone and non-monotone maps are obtained as special cases.

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Section: Introductionmentioning

confidence: 90%

“…Colebunders et al 2011). The object ([0, ∞], q σ ) is initially dense in App, meaning that for every approach space X, the total source (g : X → ([0, ∞], q σ )) g contraction in App is initial in App.…”

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confidence: 99%

Fixed points of contractive maps on dcpo's

Colebunders

Wachter

Löwen

2013

Math. Struct. Comp. Sci.

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“…To address this issue, approach spaces have been introduced by Robert Lowen [18] which are based upon point to set distance. Since approach spaces are the generalization of metric and topological spaces, several applications in different areas of mathematics including probability theory [15], domain theory [16], group theory [20] and vector spaces [21] naturally exist. However, App (category of approach spaces and contraction maps) fails to enjoy some convenience categorical properties such as cartesian closedness and hereditary properties.…”

Section: Introductionmentioning

confidence: 99%

Closure operators in convergence approach spaces

Qasim¹,

Baran²,

Abughalwa³

2021

Turk J Math

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In this paper, we characterize closed and strongly closed subsets of convergence approach spaces and introduce two notions of closure in the category of convergence approach spaces which satisfy idempotent, productive and (weakly) hereditary properties. Furthermore, we explicitly characterize each of Ti convergence-approach spaces, i = 0, 1, 2 with respect to these closure operators and show that each of these subcategories of Ti convergence approach spaces, i = 0, 1, 2 are epireflective as well as we investigate the relationship among these subcategories. Finally, we characterize connected convergence approach spaces.

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“…Approach spaces have been introduced by Lowen [30,31] to generalize metric and topological concepts, have many applications in almost all areas of mathematics including probability theory [15], convergence theory [16], domain theory [17], and fixed point theory [18]. Due to its huge importance, several generalizations of approach spaces appeared recently such as probabilistic approach spaces [22], quantale-valued gauge spaces [23], quantale-valued approach system [24], and quantale-valued approach with respect to (w.r.t.)…”

Section: Introductionmentioning

confidence: 99%