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Diego Moya (talk | contribs) specific name that is still ambiguous |
m Correct slight error in definition of tree corresponding to a closed subset of <math>X^\omega<\math>. |
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A node (that is, element) of <math>T</math> is '''terminal''' if there is no node of <math>T</math> properly extending it; that is, <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 with no terminal nodes is called '''pruned'''.
If we equip <math>X^\omega</math> with the [[product topology]] (treating ''X'' as a [[discrete space]]), then every closed subset <math>C</math> of
<math>X^\omega</math> is of the form <math>[T]</math> for some pruned tree <math>T</math> (namely, <math>T:= \{ \vec x|n: n \in \omega, x\in
Frequently trees on [[cartesian product]]s <math>X\times Y</math> are considered. In this case, by convention, the set <math>(X\times Y)^{\omega}</math> is identified in the natural way with a subset of <math>X^{\omega}\times Y^{\omega}</math>, and <math>[T]</math> is considered as a subset of <math>X^{\omega}\times Y^{\omega}</math>. We may then form the '''projection''' of <math>[T]</math>,
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