IJPAM: Volume 68, No. 3 (2011)
A CLOSED SET BY ONE CONTINUOUS OFF A POLYHEDRON
Department of Psychiatry
College of Physicians and Surgeons
1051, Riverside Drive, P.O. Box 42, New York, NY 10032, USA
Abstract. Let be a finite simplicial complex (i.e., a finite collection of simplices that fit together nicely) with underlying space (union of simplices in ) . Let be a subcomplex of . Let . Then there exists , depending only on and , with the following property. Let be closed and suppose is a continuous map of into some topological space (``'' indicates set-theoretic subtraction). Suppose , where ``'' indicates Hausdorff dimension. Then there exists such that is the underlying space of a subcomplex of and there is a continuous map of into such that:
- , where denotes -dimensional Hausdorff measure;
- if then belongs to a simplex in intersecting ;
- if , , and does not intersect any simplex in whose simplicial interior intersects , then is defined and equals ;
- if then ;
- if is a metric space and is locally Lipschitz on then is locally Lipschitz on ; and
- and .
Moreover, can be replaced by an arbitrarily fine subdivision without changing . Consequently, modulo subdivision, if , we may assume if and we may assume .
Note that can be any closed subset of . For example, no rectifiability assumptions on are required.
Received: December 23, 2010
AMS Subject Classification: 28A75, 51M20
Key Words and Phrases: simplicial complex, deformation theorem
Source: International Journal of Pure and Applied Mathematics