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A technique often used in
topology involves the inductive modification of a given mapping in order to achieve a
limit mapping having certain prescribed properties. The following definition
will facilitate the discussion. Suppose X and Y are topological spaces, and
{Wi},i = 1,2,⋯ , is a countable collection of subsets of X. Then a sequence
{fi},i ≧ 0, of mappings from X into Y is called stable relative to {Wi} if
fi|(X − Wi) = fi−1|(X − Wẋ),i,= 1,2,⋯ . Note, in the above definition, that if
{Wi} is a locally finite collection, then limi→∞fi is necessarily a well defined
mapping from X into Y , and is continuous if each fi is continuous. In a typical
smoothing theorem, a Cr-mapping f : M → N between C∞ differentiable manifolds
M and N is approximated by a C∞-mapping g : M → N, where the mapping g is
constructed as the limit of a suitable sequence {fi}(with f0 = f) which is stable
relative to a locally finite collection {Ci} of compact subsets of M. On the other
hand, instead of improving f, it is also of interest to approximate f by a mapping g
which has bad behavior at, say, a dense set of points of M. In this paper, such a
mapping g is constructed as the limit of a sequence {fi} (with f0 = f) which is
stable relative to {Ci}, but where the Ci are more “clustered” than a locally finite
collection. The case of interest here is where a sequence of homeomorphisms {Hi},
which is stable relative to {Ui}, necessarily converges to a homeomorphism.
Theorem 1 of this paper gives a sufficient condition that the latter be satisfied
for homeomorphisms of a metric space. In Theorem 1, the collection {Ui}
is not, in general, locally finite (in fact, the Ui satisfy a certain “nested”
condition). Theorem 1 is used to establish a result concerning the distribution of
homeomorphisms (of a differentiable manifold) which have a dense set of spiral
points.
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