Abstract:
The classical dimension theories of Menger-Urysohn and Lebesgue are equivalent on metric spaces. However, their infinite-dimensional analogues may differ, even on compact metric spaces. The three such infinite-dimensional dimension theories considered in this thesis are known as countable-dimensionality, property C, and weak infinite-dimensionality. The open questions regarding the relationships between these properties forms what is called "the Generalized Alexandroff Question". In the second chapter of this thesis, results are obtained regarding the types of infinite-dimensional spaces involved with the use of various classes of maps. Inclusion maps, projection maps onto factors of product spaces, open mappings, refinable mappings, and approximately invertible maps such as hereditary shape equivalences are investigated. In the final chapter of this thesis, a characterization of weak infinite-dimensionality in terms of binary open covers is generalized to give an infinite number of such characterizations. These will be used to better understand the differences between infinite-dimensional dimension theories.
