Tree (descriptive set theory): Difference between revisions

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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}x_n\rangle</math> is a sequence of length <math>n</math> in <math>T</math>, and if <math>0\le m<n</math>,

then the shortened sequence <math>\langle x_0,x_1,\ldots,x_{m-1}x_m\rangle</math> also belongs to <math>T</math>.  In particular, choosing <math>m=0</math> shows that the empty sequence belongs to every tree.

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