Generalized Cantor manifolds and homogeneity

Details Generalized Cantor manifolds and homogeneity Generalized Cantor manifolds and homogeneity

2008-07-23

arxiv.org/abs/0807.3756v1

Mathematics

General Topology

26 pages, 3 figures

Scientific paper

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.

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