A topological space is
called locally finite-dimensional if every point has a neighborhood of finite
(covering) dimensiondim. In the class of metric spaces, it is shown that every
locally finite-dimensional space has small inductive dimension ind ≦ ω, and is
strongly countable-dimensional (hence is also countable-dimensional). Examples
are given to shown that the converses of these statements are false, and
that the property of being locally finite-dimensional neither implies nor is
implied by that of having large inductive dimension Ind. Sum theorems are
included, of which the following is representative: a metric space is strongly
countable-dimensional iff it is the union of a locally countable collection of closed
finite-dimensional, locally finite-dimensional, or strongly countable-dimensional
subsets.
