Extended topology: connected sets and Wallace separations
Hammer, Preston C. (1963)
Portugaliae mathematica
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Hammer, Preston C. (1963)
Portugaliae mathematica
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Guido, Cosimo
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Rudolph Hrycay (1970)
Fundamenta Mathematicae
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J. Guthrie, H. Stone (1973)
Fundamenta Mathematicae
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J. Hunt (1980)
Fundamenta Mathematicae
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R. Moore (1922)
Fundamenta Mathematicae
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Sierpinski has shown (Wacław Sierpiński Sur une condition pour qu'un continu soit une courbe jordanienne, Fundamenta Mathematicae I (1920), pp. 44-60) that in order that a closed and connected set of points M should be a continuous curve it is necessary and sufficient that, for every positive number ϵ, the connected point-set M should be the sum of a finite number of closed and connected point-sets each of diameter less than ϵ. It follows that, as applied to point-sets which are closed,...
Charatonik, J.J. (1996)
Portugaliae Mathematica
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Yatsuka Nakamura (2006)
Formalized Mathematics
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First, equivalence conditions for connectedness are examined for a finite topological space (originated in [9]). Secondly, definitions of subspace, and components of the subspace of a finite topological space are given. Lastly, concepts of continuous finite sequence and minimum path of finite topological space are proposed.
Coloma Ballester, Vicent Caselles (2001)
Publicacions Matemàtiques
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We prove that the topographic map structure of upper semicontinuous functions, defined in terms of classical connected components of its level sets, and of functions of bounded variation (or a generalization, the WBV functions), defined in terms of M-connected components of its level sets, coincides when the function is a continuous function in WBV. Both function spaces are frequently used as models for images. Thus, if the domain Ω' of the image is Jordan domain, a rectangle, for instance,...