Vol. 42, No. 1, 1972

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Finite-dimensional properties of infinite-dimensional spaces

B. R. Wenner

Vol. 42 (1972), No. 1, 267–276

A topological space is called locally finite-dimensional if every point has a neighborhood of finite (covering) dimensiondim. In the class of metric spaces, it is shown that every locally finite-dimensional space has small inductive dimension ind ω, and is strongly countable-dimensional (hence is also countable-dimensional). Examples are given to shown that the converses of these statements are false, and that the property of being locally finite-dimensional neither implies nor is implied by that of having large inductive dimension Ind. Sum theorems are included, of which the following is representative: a metric space is strongly countable-dimensional iff it is the union of a locally countable collection of closed finite-dimensional, locally finite-dimensional, or strongly countable-dimensional subsets.

Mathematical Subject Classification 2000
Primary: 54F45
Received: 19 April 1971
Published: 1 July 1972
B. R. Wenner