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One says a topological subspace is connected if it is so for the induced topology. $\mathbf{Lemma 4.4}$ Let X be a topological space and A ⊂ X an open and closed subset. For any connected subspace Y ⊂ X either Y ⊂ A or Y ∩ A = ∅. Proof The intersection Y ∩ A is open and closed in Y . As Y is connected, necessarily Y ∩ A = Y (hence Y ⊂ A) or Y ∩ A = ∅.
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Mon, 24 Apr 2023 15:13 GMT