Vol. 95, No. 2, 1981

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𝜀-covering dimension

Allan Calder, William H. Julian, Ray Mines, III and Fred Richman

Vol. 95 (1981), No. 2, 257–262

A compact metric space T has Lebesgue covering dimension at most n if for each positive 𝜀 the space T has an 𝜀-cover of order at most n. We show that if T is a compact subset of Euclidean n-space and T has an 𝜀-cover of order at most n 2, then any two points whose distance from T is greater than 𝜀 can be joined by a path bounded away from T. This refines, and provides a constructive proof for, the theorem that the complement of an (n 2)-dimensional compact subset of Euclidean n-space is connected.

Mathematical Subject Classification 2000
Primary: 54F45
Received: 20 June 1980
Published: 1 August 1981
Allan Calder
William H. Julian
Department of Mathematical Sciences
New Mexico State University
Las Cruces NM 88003
United States
Ray Mines, III
Fred Richman