Connection Structures: Grzegorczyk's and Whitehead's Definitions of Point

Biacino, Loredana; Gerla, Giangiacomo

doi:10.1305/ndjfl/1039886519

Cited by 38 publications

(58 citation statements)

References 6 publications

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“…Remark 2.2. Biacino and Gerla [2] interpret the term 'mereological field' as "the structure obtained from a complete Boolean algebra B by deleting the zero-element, i.e., R = B − {0}" (p. 432). Therefore, their mereological fields are mereological structures, i.e., our complete mereological fields (cf.…”

Section: Mereological Structuresmentioning

confidence: 99%

“…Moreover, the axioms are either postulates or theorems of theories which are known as some standard approaches to the problem in the literature (see e.g. [1,2,7,20,24]). …”

Section: Separation Structuresmentioning

confidence: 99%

“…15 Biacino and Gerla [2] state this fact without proof (as Theorem 3.3) unnecessarily requiring that T be a Hausdorff space. Moreover, by Proposition 4.3, if T is weakly regular, then qsepT belongs to Sep c and so it will be called the separation structure associated with T ; so we write: sepT .…”

Section: [ Satisfies (Mf) and (S1)-(s3) For (S4): For All A B C ∈ mentioning

confidence: 99%

“…These are, in a given quasi-separation structure, all filters generated by pre-points. 16 16 We characterize the notion of point in quasi-separation structures similarly to Biacino and Gerla [2]. Grzegorczyk Let R = R, )( be any quasi-separation structure.…”

Section: Representatives Of Points and Points In Quasi-separation Strmentioning

confidence: 99%

“…435-436]. The differences lie in two facts: in [2] the counterparts of the theorem were proven after the notion of point had been introduced, and different spaces were taken into account. Proof.…”

Section: Grzegorczyk Structures Versus Separation Structuresmentioning

confidence: 99%

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A Study in Grzegorczyk Point-Free Topology Part I: Separation and Grzegorczyk Structures

Gruszczyński

Pietruszczak

2018

Stud Logica

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Abstract.This is the first, out of two papers, devoted to Andrzej Grzegorczyk's pointfree system of topology from Grzegorczyk (Synthese 12(2-3): [228][229][230][231][232][233][234][235] 1960. https://doi. org/10.1007/BF00485101). His system was one of the very first fully fledged axiomatizations of topology based on the notions of region, parthood and separation (the dual notion of connection). Its peculiar and interesting feature is the definition of point, whose intention is to grasp our geometrical intuitions of points as systems of shrinking regions of space. In this part we analyze (quasi-)separation structures and Grzegorczyk structures, and establish their properties which will be useful in the sequel. We prove that in the class of Urysohn spaces with countable chain condition, to every topologically interpreted representative of a point in the sense of Grzegorczyk's corresponds exactly one point of a space. We also demonstrate that Tychonoff first-countable spaces give rise to complete Grzegorczyk structures. The results established below will be used in the second part devoted to points and topological spaces.

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Section: Mereological Structuresmentioning

confidence: 99%

“…Moreover, the axioms are either postulates or theorems of theories which are known as some standard approaches to the problem in the literature (see e.g. [1,2,7,20,24]). …”

Section: Separation Structuresmentioning

confidence: 99%

Section: [ Satisfies (Mf) and (S1)-(s3) For (S4): For All A B C ∈ mentioning

confidence: 99%

Section: Representatives Of Points and Points In Quasi-separation Strmentioning

confidence: 99%

Section: Grzegorczyk Structures Versus Separation Structuresmentioning

confidence: 99%

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