5 Addition and Subspace Theorems for Asymptotic Large Inductive Dimension

We prove addition and subspace theorems for asymptotic large inductive dimension. We investigate a transfinite extension of this dimension and show that it is trivial. 0. Asymptotic dimension asdim of a metric space was defined by Gromov for studying asymptotic invariants of discrete groups [1]. This dimension can be considered as asymptotic analogue of the Lebesgue covering dimension dim. Dranishnikov has introduced dimension asInd which is analogous to large inductive dimension Ind [2]. Some of basic theorems of classical dimension theory are sum, addition and subspace theorems for different dimensions and classes of topological spaces. Here we mention some of them related to dimension Ind. (All mentioned facts from classic dimension theory could be found in [3]).