König's theorem is an important result in set theory and combinatorial set theory, specifically related to the study of infinite trees. The theorem states the following: If \( T \) is an infinite tree of finite height such that every node in \( T \) has a finite number of children, then \( T \) has either: 1. An infinite branch (a path through the tree that visits infinitely many nodes), or 2.

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If for every , then
The injection uses functions supported at one coordinate. For non-surjectivity, given any map from the disjoint union to the product, choose at coordinate a value omitted by the th row; the resulting diagonal element is outside its image.