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A.836

Part of KoMaL A Problems 2022/2023

Problems(1)

Compactness property with partitions of N

Source: KoMaL A. 836

12/13/2022
For every i∈Ni \in \mathbb{N} let AiA_i, BiB_i and CiC_i be three finite and pairwise disjoint subsets of N\mathbb{N}. Suppose that for every pairwise disjoint sets AA, BB and C C with union N\mathbb N there exists i∈Ni\in \mathbb{N} such that Ai⊂AA_i \subset A, Bi⊂BB_i \subset B and Ci⊂CC_i \subset C. Prove that there also exists a finite S⊂NS\subset \mathbb{N} such that for every pairwise disjoint sets AA, BB and CC with union N\mathbb N there exists i∈Si\in S such that Ai⊂AA_i \subset A, Bi⊂BB_i \subset B and Ci⊂CC_i \subset C.
Submitted by András Imolay, Budapest
combinatoricstopologykomal