User:MFH/Branch (graph theory)

Branch (graph theory)
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In mathematics, especially mathematical logic and set theory, a branch of a tree T is a map f such that ∀n : f (n) ∈ T.

By a tree on a product set κ1 ×···× κp we mean here a subset of the union of κ1i×···×κpi for all i < ω,

$$   T\subseteq\bigcup_{i<\omega} \kappa_1^i\times\cdots\times\kappa_p^i ~,$$

closed under initial segments, and the set of branches of such a tree is then more explicitly the set

$$   [T]=\{ (f_1,\dots,f_p)\mid\forall n\in\omega: (f_1(n),\dots,f_p(n))\in T\} ~.$$

This is a closed set for the usual product topology (see AD plus).

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