The dimension of the kernel of a planar set

Marilyn Breen

Vol. 82 (1979), No. 1, 15–21

DOI: 10.2140/pjm.1979.82.15

Abstract

Let S be a compact subset of R². We establish the following: For 1 ≦ k ≦ 2, the dimension of ker S is at least k if and only if for some 𝜖 > 0, every f(k) points of S see via S a common k-dimensional neighborhood having radius 𝜖, where f(1) = 4 and f(2) = 3. The number f(k) in the theorem is best possible.

Mathematical Subject Classification 2000

Primary: 52A30

Secondary: 52A35

Milestones

Received: 15 January 1978

Published: 1 May 1979