Tree (descriptive set theory): Difference between revisions

Revision as of 22:00, 27 February 2017

Revision as of 22:00, 27 February 2017

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===Trees===

The collection of all finite sequences of elements of a set <math>X</math> is denoted <math>X^{<\omega}</math>.

With this notation, a tree is a nonempty subset <math>T</math> of <math>X^{<\omega}</math>, such that if

<math>\langle x_0,x_1,\ldots,x_{n-1}\rangle</math> is a sequence of length <math>n</math> in <math>T</math>, and if <math>0\le m<n</math>,

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===Terminal nodes===

A finite sequence that belongs to a tree <math>T</math> is called a '''terminal node''' if it is not a prefix of a longer sequence in <math>T</math>. Equivalently, <math>\langle x_0,x_1,\ldots,x_{n-1}\rangle \in T</math> is terminal if there is no element <math>x</math> of <math>X</math> such that that <math>\langle x_0,x_1,\ldots,x_{n-1},x\rangle \in T</math>. A tree that does not have any terminal nodes is called '''pruned''' This is all bullshit.

==Relation to other types of trees==