Generalized Cantor manifolds and homogeneity

Abstract

A classical theorem of Alexandroff states that every $n$-dimensional compactum $X$ contains an $n$-dimensional Cantor manifold. This theorem has a number of generalizations obtained by various authors. We consider extension-dimensional and infinite dimensional analogs of strong Cantor manifolds, Mazurkiewicz manifolds, and $V^n$-continua, and prove corresponding versions of the above theorem. We apply our results to show that each homogeneous metrizable continuum which is not in a given class $\mathcal{C}$ is a strong Cantor manifold (or at least a Cantor manifold) with respect to $\mathcal{C}$. Here, the class $\mathcal{C}$ is one of four classes that are defined in terms of dimension-like invariants. A class of spaces having bases of neighborhoods satisfying certain special conditions is also considered.

26 pages, 3 figures

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• Mazurkiewicz manifolds and homogeneity