IJPAM: Volume 68, No. 3 (2011)
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: spe4@columbia.edu
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