Mathematics · Topology

Let X be a non-empty set. Suppose that τ 1 and τ 2 are two topologies over X such that τ 2 ⊂ τ 1 . Which of the following statements implies that τ 1 = τ 2 ?

Verified Solution4 OptionsUpdated: October 2026👁️1 View

1For any subset A ⊂ X, the closure of A in (X, τ 2 ) is contained in the closure of A in (X, τ 1 )
2The connected components of both (X, τ 1 ) and (X, τ 2 ) are the same
3(X, τ 1 ) is compact and τ 2 is T 2 (Hausdorff)✓
4(X, τ 1 ) is compact and τ 1 is T 2 (Hausdorff)

Correct answer

(X, τ 1 ) is compact and τ 2 is T 2 (Hausdorff)

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Let X be a non-empty set. Suppose that &tau; 1 and &tau; 2 are two topologies over X such that &tau; 2 &sub; &tau; 1 . Which of the following statements implies that &tau; 1 = &tau; 2 ?