Vol. 27, No. 1, 1968

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Dimension on boundaries of 𝜀-spheres

B. R. Wenner

Vol. 27 (1968), No. 1, 201–210

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.

Mathematical Subject Classification
Primary: 54.35
Received: 1 August 1967
Published: 1 October 1968
B. R. Wenner