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...
