# IJPAM: Volume 68, No. 3 (2011)

**APPROXIMATING A FUNCTION CONTINUOUS OFF**

A CLOSED SET BY ONE CONTINUOUS OFF A POLYHEDRON

A CLOSED SET BY ONE CONTINUOUS OFF A POLYHEDRON

Department of Psychiatry

College of Physicians and Surgeons

Columbia University

1051, Riverside Drive, P.O. Box 42, New York, NY 10032, USA

e-mail: [email protected]

**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

**ISSN:** 1311-8080

**Year:** 2011

**Volume:** 68

**Issue:** 3