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Sequence of subsets of Hausdorff space

Source: Miklos Schweitzer 2023, Problem 2

March 7, 2024
hausdorff spacetopologycollege contestsMiklos Schweitzer

Problem Statement

Let G0,G1,…G_0, G_1,\ldots be infinite open subsets of a Hausdorff space. Prove that there exist some infinite pairwise disjoint open sets V0,V1,…V_0,V_1,\ldots and some indices n0<n1<⋯n_0<n_1<\cdots such that Vi⊆GniV_i\subseteq G_{n_i} for every i⩾0.i\geqslant 0.
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