An Extension of Hall's Theorem

Abstract

Let \( (A_i) _{i \in I} \) and \( (B_i) _ {i \in I} \) be two (possibly infinite) families of finite sets. Let cl(P) denote the closure of the set \( P := \{ ({A_i}, {B_i} ): i \in I \} \) of the pairs with respect to the componentwise union and intersection operations. Then there exists an injective map \( {\displaystyle \bigcup _ {i \in I}} A_i \rightarrow {\displaystyle \bigcup _ {i \in I }} B_i \) such that \( f (A_i) \subseteq B_i \) for every i if, and only if, card \( (A) \leq \) card (B) for every pair \( (A, B) \in cl (P) \).