Let X and X' denote a single set in the two topologies T and T', respectively. Let i : X' → X be the identity function. (a) Show that i is continuous A T' is finer than T. (b) Show that i is a homeomorphism + T' = T.

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A: Given below the detailed solution

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A: No information is given for topology on Y So i have solved only the middle question

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A: Step 1: Step 2: Step 3: Step 4:

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A: Step 1: Step 2: Step 3: Step 4:

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A: Given statement is If T:X→Y is continuous, linear and open, then T is surjective.

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Q: 4. Let A := (0, 1] and let f : A → R be defined by f(x) = !. Prove that f is continuous on A.

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