## Effectively Closed Sets and Enumerations (2007)

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### BibTeX

@MISC{Brodhead07effectivelyclosed,

author = {Paul Brodhead and Douglas Cenzer},

title = {Effectively Closed Sets and Enumerations},

year = {2007}

}

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### Abstract

An effectively closed set, or Π 0 1 class, may viewed as the set of infinite paths through a computable tree. A numbering, or enumeration, is a map from ω onto a countable collection of objects. One numbering is reducible to another if equality holds after the second is composed with a computable function. Many commonly used numberings of Π 0 1 classes are shown to be mutually reducible via a computable permutation. Computable injective numberings are given for the family of Π 0 1 classes and for the subclasses of decidable and of homogeneous Π 0 1 classes. However no computable numberings exist for small or thin classes. No computable numbering of trees exists that includes all computable trees without dead ends.