Vol. 24, No. 1, 1968

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Sequences of homeomorphisms which converge to homeomorphisms

Jerome L. Paul

Vol. 24 (1968), No. 1, 143–152

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) = fi1|(X W),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.

Mathematical Subject Classification
Primary: 54.60
Received: 28 December 1966
Published: 1 January 1968
Jerome L. Paul