On the sum of the Alexandroff's manifolds
by
Vladimir Todorov
University of Architecture and Civil Engineering, Sofia, Bulgaria

In 1957 [Die Kontinua (V^p)-eine Versharfung der Cantorshen Mannigfaktigkeiten, Monatsh. fur Math und Phys., 61, 1, 67-76] P. Alexandroff has introduced the so-called continua (V^p), a stronger class of Cantor manifolds that have all the properties of the classical Cantor manifolds.

More precisely, the compactum X is defined to be a continuum (V^p) if dimX=p and for each disjoint pair U, V of nonempty open subsets of X there exists a positive e such that for every partition C between U and V for the (p-2) - dimesional diameter a^p-2C of C we have a^p-2(C) ³ e.

In his paper Alexandroff gave a hypothesis concerning the sum of two continua (V^p):

If by the kernel of condensation of a given p-dimensional compactum X one understand each compact subspace X₀, which is contained in some nonwhere dense p-dimesional compactum X¢ Ì X, then the sum XÈY of the (V^p) continua X and Y should be a (V^p) continuum if and only if the itersection XÇY does not contain kernels of condensation (in X and in Y in the same time).

It is shown in this discussion that the Alexandroff's conjecture is neither a necessary, nor a sufficient condition. A necessary and sufficient condition for the Alexandroff's hypothesis is also given. We discuss as well the problem of existance of dimensional components in the sense of continua (V^p).

Date received: January 26, 1998