Mathematics · Topology

Let Y be a nonempty bounded, open subset of Rn and let Y̅ denote its closure. Let {Uj}j ≥ 1 be a collection of open sets in Rn such that Y̅ ⊆ ∪j ≥ 1 Uj. Which of the following statements is false?

Verified Solution4 OptionsUpdated: October 2026👁️1 View

1There exists a subsequence j1, j2, ... such that Y ⊆ ∪k=1∞ Ujk
2For every subsequence j1, j2, ... we have Y ⊆ ∪k=1∞ Ujk
3There exists a positive integer N such that Y ⊆ ∪j=1N Uj✓
4There exist finitely many positive integers j1, ..., jN such that Y ⊆ ∪k=1N Ujk

Correct answer

There exists a positive integer N such that Y ⊆ ∪j=1N Uj

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