Continuum theory. II.

Abstract

This work is a continuation of the survey "Continuum theory. I," by the same authors. Here we discuss results related to the notion of aposyndcsis, dimension theory, and the Hahn-Mazurkicwicz problem.* APOSYNDETIC CONTINUAIn 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 of each complementary domain is locally connected.From the duality between the notions of aposyndesis and semi-local connectivity it follows that, being pointwise different, these notions are identical globally, i.e., every semi-locally connected continuum is aposyndetic at any of its points, and vice versa.What continua are aposyndetic? Each product of two or more non-degenerate continua is aposyndetic. Moreover, FitzGerald proved [35] that a product of two nontrivial continua is finitely aposyndetic, i.e., n-point aposyndetic for every positive integer n. The latter means that a continuum is aposyndetic at each of its points with respect to n arbitrary points, noncoinciding with the given point. Every aposyndetic continuum is decomposable. Hence, in particular, the Cech-Stone remainder of the half-line is not aposyndetic. On the other hand, Bellamy proved [2] that the Cech-Stone remainder of the Euclidean space R", n _> 2, is aposyndetic. But none of these remainders is locally connected. Th...

The topological fixed point property –- an elementary continuum-theoretic approach

“…Despite of the positive results (1), (2), (3), (5), (7), (8), (10), (11), (13), (18) and (19), one could say the FPP is "elusive", which means "difficult to catch" (cf. the Longman Dictionary of Contemporary English, 1987).…”