Vol. 26, No. 1, 1968

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D-dimension. I. A new transfinite dimension

David Wilson Henderson

Vol. 26 (1968), No. 1, 91–107

The large inductive dimension (Ind) can be extended, by transfinite induction, to all the ordinals. The transfinite inductive dimension so obtained has been investigated by many authors. Unfortunately, it does not possess many of the nice properties which are possessed by the inductive dimension in finite-dimensional spaces. For instance, the transfinite inductive dimension fails to be monotone for separable metric spaces; and it fails to satisfy the sum theorem, even for compact metric spaces.

This paper introduces a new transfinite dimension called D-dimension, which is defined for all metric spaces. For finite-dimensional spaces, D-dimension equals Ind. It is shown that D-dimension is also a monotone and local property and that it satisfies several sum and product theorems. These properties lead to a characterization of D-dimension.

Mathematical Subject Classification
Primary: 54.70
Received: 26 July 1967
Revised: 25 October 1967
Published: 1 July 1968
David Wilson Henderson