Vol. 26, No. 1, 1968

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D-dimension. II. Separable spaces and compactifications

David Wilson Henderson

Vol. 26 (1968), No. 1, 109–113

This paper continues the discussion of a new transfinite dimension which was introduced by the author in “D-dimension, I. A new transfinite dimension,” Pacific J. Math. vol. 26. In the first part of this paper we show that, for a metric space X, D(X) is an ordinal if and only if each closed subset Y X contains a dense open (in Y ) subset each of whose points has a finite-dimensional neighborhood. It follows that if X is complete and separable, then X is weakly countable-dimensional (i.e. the union of a countable number of closed finite-dimensional subsets) if and only if D(X) is an ordinal. It is also shown that, if Ind(X) exists, then Ind(X) < D(X); furthermore, if X is compact and Ind(X) does not exists, then D(X) is not an ordinal. In the second part, it is proved that each weakly infinite-dimensional separable metric space has a compactification with the same D-dimension; an example is given to show that this is not true for all separable metric spaces.

Mathematical Subject Classification
Primary: 54.70
Received: 1 August 1967
Published: 1 July 1968
David Wilson Henderson