On a Problem of F. Riesz Concerning Proximity Structures

Thron, W. J.

doi:10.2307/2038689

Cited by 7 publications

(3 citation statements)

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“…This similarity was responsible for the late reception of Riesz's topological ideas, e.g. Thron (1973). 73 That is how the Rome lecture is commonly referred to, e.g.…”

Section: Receptionmentioning

confidence: 99%

Frigyes Riesz and the emergence of general topology

Rodríguez

2014

Arch. Hist. Exact Sci.

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In 1906, Frigyes Riesz introduced a preliminary version of the notion of a topological space. He called it a mathematical continuum. This development can be traced back to the end of 1904 when, genuinely interested in taking up Hilbert's foundations of geometry from 1902, Riesz aimed to extend Hilbert's notion of a two-dimensional manifold to the three-dimensional case. Starting with the plane as an abstract point-set, Hilbert had postulated the existence of a system of neighbourhoods, thereby introducing the notion of an accumulation point for the point-sets of the plane. Inspired by Hilbert's technical approach, as well as by recent developments in analysis and point-set topology in France, Riesz defined the concept of a mathematical continuum as an abstract set provided with a notion of an accumulation point. In addition, he developed further elementary concepts in abstract point-set topology. Taking an abstract topological approach, he formulated a concept of three-dimensional continuous space that resembles the modern concept of a three-dimensional topological manifold. In 1908, Riesz presented his concept of mathematical continuum at the International Congress of Mathematicians in Rome. His lecture immediately won the attention of people interested in carrying on his research. They promoted his ideas, thus assuring their gradual reception by several future founders of general topology. In this way, Riesz's work contributed significantly to the emergence of this discipline.

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“…This similarity was responsible for the late reception of Riesz's topological ideas, e.g. Thron (1973). 73 That is how the Rome lecture is commonly referred to, e.g.…”

Section: Receptionmentioning

confidence: 99%

Frigyes Riesz and the emergence of general topology

Rodríguez

2014

Arch. Hist. Exact Sci.

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“…It is known that if AδB for two subsets A and B of the proximity space (X, δ) then there exists a cluster C in (X, δ) such that A, B ∈ C ( [14]). But this result can not be generalized to all nearness spaces.…”

Section: Nearness Structures Induced From Large Scale Resemblancesmentioning

confidence: 99%

A concept for resemblance in large scale geometry

Kalantari¹

2023

Preprint

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In this paper, we introduce the notion of large scale resemblance structure as a new large scale structure by axiomatizing the concept of being alike in large scale for a family of subsets of a set. We see that in a particular case, large scale resemblances on a set can induce a nearness on it, and as a consequence, we offer a relatively big class of examples to show that not every near family is contained in a bunch. Besides, We show how some large scale properties like asymptotic dimension can be generalized to large scale resemblance spaces.

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“…Several attempts have been made to generalize these results for some proximity relations weaker than that of Efremovic. Lodato [5], Gagrat and Naimpally [2], and Thron [10] worked with proximities compatible with /? 0 -spaces, while Harris [3] tried with proximities compatible with those regular spaces which can be densely embedded in regular closed spaces.…”

Section: Introductionmentioning

confidence: 99%