Vol. 38, No. 1, 1971

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Examples concerning sum properties for metric-dependent dimension functions

John Charles Nichols and James C. Smith

Vol. 38 (1971), No. 1, 151–159

Let d0 denote the metric dimension function defined by Katětov, and let dim be the covering dimension function. K. Nagami and J. H. Roberts introduced the metric-dependent dimension functions d2 and d3, and J. C. Smith defined the functions d6 and d7. The following relations hold for all metric spaces (X,ρ):

d2(X,ρ) ≦ da(X, ρ) ≦ d6(X,ρ) ≦ d7(X,ρ) ≦ d0(X, ρ).

Since all of the metric-dependent dimension functions above satisfy a “Weak Sum Theorem,” it is natural to ask if any of these functions satisfy the Finite Sum Theorem or the Countable Sum Theorem. In this paper the authors obtain new properties of these dimension functions, and using these results construct examples for which none of the metric dependent dimension functions satisfy either of the sum theorems in question.

Mathematical Subject Classification 2000
Primary: 54F45
Received: 2 March 1971
Published: 1 July 1971
John Charles Nichols
James C. Smith