The purpose of this paper is
to make somewhat more accessible the topological dimension-theoretic properties of
metric spaces. We shall show that any metric for a space can be replaced
by a topologically equivalent metric which has the following property: the
boundary of any 𝜖-sphere meets each of a specified countable collection of closed,
finite-dimensional subsets in a set of lower dimension. An additional property
of the new metric is that for any fixed 𝜖, the collection of all 𝜖-spheres is
closure-preserving.
In the case of a separable metric space, the result can be sharpened to produce a
totally bounded metric with the above properties, and in this case we obtain for each
fixed 𝜖 at most finitely many distinct 𝜖-spheres.
