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1

Digital Jordan Curves and Surfaces with Respect to a Closure Operator

Šlapal Josef

Fundamenta Informaticae

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2021

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Vol. 179, nr 1

59--74

EN

In this paper, we propose new definitions of digital Jordan curves and digital Jordan surfaces. We start with introducing and studying closure operators on a given set that are associated with n-ary relations (n > 1 an integer) on this set. Discussed are in particular the closure operators associated with certain n-ary relations on the digital line ℤ. Of these relations, we focus on a ternary one equipping the digital plane ℤ2 and the digital space ℤ3 with the closure operator associated with the direct product of two and three, respectively, copies of this ternary relation. The connectedness provided by the closure operator is shown to be suitable for defining digital curves satisfying a digital Jordan curve theorem and digital surfaces satisfying a digital Jordan surface theorem.

2

The notion of connectedness in mathematical analysis of XIX century

Sinkevich G. I.

Czasopismo Techniczne. Nauki Podstawowe

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2014

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R. 111, z. 1-NP

195--209

EN

The notion of connectedness was introduced by Listing in 1847 and was further developed by Riemann, Jordan and Poincaré. The notion and rigorous definition of metric and topological space were formed in Frechet’s works in 1906, and in Hausdorff’s works in 1914. The notion of continuum could be traced back to antiquity, but its mathematical definition was formed in XIX century, in the works of Cantor and Dedekind, later of Hausdorff and Riesz. Karl Weierstrass (1815–1897) brought mathematical analysis to a rigorous form; also, the notions of future areas of mathematics – functional analysis and topology – were formed in his reasoning. Weierstrass’s works were not translated into Russian, and his lectures were not published even in Germany. In 1989, synopses of his lectures devoted to additional chapters of the theory of functions were published. Their material served as the basis for this article.

PL

Pojęcie spójności zostało wprowadzone w 1847 roku przez Listinga, a dalej zostało opracowane przez Riemanna, Jordana i Poincarégo. Pojęcie i rygorystyczna definicja przestrzeni metrycznej i topologicznej pojawiły się w pracach Frécheta w 1906 roku i Hausdorffa w 1914 roku. Pojęcie kontinuum sięga starożytności, ale jego matematyczna definicja powstała w XIX wieku, w pracach Cantora i Dedekinda, później Hausdorffa i Riesza. Karl Weierstrass (1815–1897) przedstawił analizę matematyczną w rygorystycznej formie; również pojęcia przyszłych dziedzin matematyki – analizy funkcjonalnej i topologii – zostały zapoczątkowane w jego rozumowaniu. Prace Weierstrassa nie zostały przetłumaczone na język rosyjski, a jego wykłady nie były publikowane nawet w Niemczech. W 1989 roku ukazały się streszczenia jego wykładów poświęconych dodatkowym rozdziałom teorii funkcji. Opublikowany w nich materiał jest podstawą tego artykułu.

3

On rare s-precontinuity for multifunctions

Ekici E., Jafari S.

Demonstratio Mathematica

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2013

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Vol. 46, nr 2

395--403

EN

In this paper, we introduce and study the notions of upper and lower rarely s-precontinuous multifunctions which are a generalization of weakly s-precontinuous multifunctions due to Ekici and Park [3].

4

The connectedness of arithmetic progressions in Furstenberg's, Golomb's, and Kirch's topologies

Szczuka P.

Demonstratio Mathematica

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2010

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Vol. 43, nr 4

899-909

EN

In this paper we examine the connectedness of arithmetic progressions in the following topologies: Furstenberg's topology on the set of integers, Golomb's topology D on the set of positive integers, and Kirch's topology D′ on the set of positive integers. Immediate consequences of these studies are theorems concerning the connectedness and the locally connectedness of the topologies D and D′ proved by S. Golomb in 1959 and A. M. Kirch in 1969.