Vertically countable spheres and their wild sets

Lowell Duane Loveland

Vol. 50 (1974), No. 2, 521–529

DOI: 10.2140/pjm.1974.50.521

Abstract

A 2-sphere S in E³ is said to have vertical order n if the intersection of each vertical line with S contains no more than n points. It is shown that S ∪ IntS is a 3-cell that is locally tame from ExtS modulo a 0-dimensional set if S has vertical order 5. A subset X of E³ is said to have countable (finite) vertical order if the intersection of X with each vertical line consists of countably (finitely) many points. A 2-sphere in E^s with countable vertical order can have a wild set of dimension no larger than one.

Mathematical Subject Classification

Primary: 57A10

Milestones

Received: 21 November 1972

Published: 1 February 1974