Question

Let X be the space of all continuous functions from [0, 1] to [0, 1] equipped with the sup metric. Let Xi be the set of injective and Xs be the set of surjective elements of A and let Xis = Xi ∩ Xs. Prove or disprove: i) Xi is closed, ii) Xs is closed, iii) Xis is closed, iv) X is connected, v) X is compact.

Answer 1

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To ohovin onto wi'must Ahw frat Then Sup t Sub 0 e contrad'cth*on mt o Than Sup a contrdichlon S m o Ho) fis onto So Hene Xs i clooed lii) Xis isnot closed

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