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• W1999470601 abstract "1. APOSYNDETIC CONTINUA In this part of the paper, a continuum is a connected, metrizable, compact space. Through the whole paper a compactum is a compact Hausdorff space. The notion of aposyndetic continuum was introduced by F. B. Jones in 1941. A connected space X is called aposyndetic at a point x 6 X if, for each point y e X {z}, there exists a connected closed neighborhood of x, which does not contain y. Aposyndesis is a generalization of the notion of a connectivity im kleinen invented by R. L. Moore. A connected space X is called connected im klelnen at a point z if any neighborhood of z contains a connected closed neighborhood of x. Every continuum which is locally connected at a point x is connected im kleinen at this point, but the converse is not true. At the same time, a continuum which is connected im kleinen at each of its points is locally connected. On the other hand, aposyndesis is dual to the notion of a semi-local connectivity, introduced by Whyburn. A connected space X is said to be semi-locally connected (at a point x) if each of its points (the point x) has arbitrarily small neighborhoods whose complements consist of only a finite number of components. The duality of aposyndesis and semi-local connectivity can be illustrated by the following assertion. A connected compactum X is semi-locally connected at a point z if it ~s aposyndetic at each point y ~ x, and, conversely, X is aposyndetic at a point x if it is semi-locally connected at each point y ~ z. The invention of semi-locally connected continua was motivated by the following well-known Torhorst's theorem (1921): For every locally connected continuum in the plane, a boundary of each complementary domain (= a component of its complement) is locally connected. In 1939, Whyburn proved that, for every semi-locally connected continuum in the plane, a boundary" @default.
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• W1999470601 date "1995-03-01" @default.
• W1999470601 modified "2023-09-26" @default.
• W1999470601 title "Continuum theory. II." @default.
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• W1999470601 doi "https://doi.org/10.1007/bf02362843" @default.
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