The Monotone Mapping Problem,
Abstract
It is shown that for m = 3,4,... there is a monotone map of Euclidean n space E sup n onto itself that is not compact. This completes the monotone mapping theorem posed by G. T. Whyburn. A key lemma in the treatment shows that there is a monotone map of a cube I sup 2 onto itself such that each point inverse intersects a base I sup 2 of I sup 3. If f is a map of I sup 3 onto I sup 3 which is a homeomorphism on Int I sup 3 and takes I sup 2 homeomorphically into I sup 2, one calls f (Int I sup 2 joined to Int I sup 3) a drainage system for I sup 3. It is shown that there is a drainage system f (Int I sup 2 joined to Int I sup 3) for I sup 3 and a monotone map g of I sup 3 - f (Int I sup 2 joined to Int I sup 3) onto I sup 3 such that g is the identity on Bd I sup 3 - Int I sup 2. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Apr 01, 1971
- Accession Number
- AD0722363
Entities
People
- R. H. Bing
Organizations
- University of Wisconsin–Madison