Recursively squeezable sets are squeezable

Abstract

In work by Freedman [F2] and Freedman-Quinn [FQ] on the topology of 4-manifolds, null decompositions whose non-singleton elements are, in the terminology of [MOR], recursively starlike-equivalent sets of filtration length 1 arise and are shown to be shrinkable. The main result of [MOR] is a general theorem covering these types of decompositions. It establishes the shrinkability of null decompositions whose non-singleton elements are recursively starlike-equivalent sets whose filtration lengths have a uniform finite upper bound. That result is the inspiration for this article. Here it is shown that the hypothesis of a uniform finite upper bound on filtration lengths is unnecessary. In outline: notions of squeezable subsets and squashable subsets of a compact metric space are defined. It is observed that starlike-equivalent sets are squeezable, and that any null decomposition of a compact metric space whose non-singleton elements are squeezable is shrinkable. It is also proved that a set is squeezable if and only if it is squashable, and that every recursively squashable set is squashable. It follows that any null decomposition of a compact metric space whose non-singleton elements are recursively squeezable is shrinkable. The latter theorem has as a corollary the main result of [MOR] with the hypothesis of a uniform finite upper bound on filtration lengths removed.