D-dimension. II. Separable spaces and compactifications
David James Henderson · Pacific Journal of Mathematics · 1968
This paper continues the discussion of a new transfinite dimension which was introduced by the author in " Zλ 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 Fc 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 Xis 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 lnά(X)<D(X); furthermore, if Xis 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.1* Separable spaces* Although the theorems in this section applyto all metric spaces, the principle results (the corollaries) apply only to certain classes of separable spaces.The notation and definitions of [1] will be used.THEOREM 1.Let X be any metrizable space.Then D{X)ΦΔ if and only if each closed subset YaX contains a dense open (in Y) subset each point of which has a finite-dimensional neighborhood.