An Extension of Hall's Theorem

Let (A_i) _{i ? I} (A_i) _{i \in I} and (B_i) _{i ? I} (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 ? I } 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 è_{i ? I} A_i ? è_{i ? I} B_i {\displaystyle \bigcup _ {i \in I}} A_i \rightarrow {\displaystyle \bigcup _ {i \in I }} B_i such that f (A_i) í B_i f (A_i) \subseteq B_i for every i if, and only if, card (A) ￡ (A) \leq card (B) for every pair (A, B) ? cl (P) (A, B) \in cl (P) .