1. Let O1 and O2 be topologies on X. (1) Show that the identity map idX : (X, O1) → (X, O2) is continuous if O2 ⊂ O1. (2) Show that the identity map idX : (X, O1) → (X, O2) is not continuous if O1⊂≠O2.

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1. Let O1 and O2 be topologies on X.

(1) Show that the identity map idX : (X, O1) → (X, O2) is continuous if O2 ⊂ O1.

(2) Show that the identity map idX : (X, O1) → (X, O2) is not continuous if O1⊂≠O2.

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