3. (Contractible spaces.) (i) A space X is said to be contractible if the identity map...

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3. (Contractible spaces.) (i) A space X is said to be contractible if the identity map Idx is homotopic to the constant map at some point of X. Prove the following. (a) X is contractible if and only if it is homotopy equivalent to a space consisting of a single point. (b) If X is contractible then it is simply connected¹. (c) If f and g are maps from Y into a contractible space X, then f~g. (ii) Recall C(X), the cone on X, is the space obtained from X × I by identifying X × {0} to a point. Prove that C(X) is contractible. 3. (Contractible spaces.) (i) A space X is said to be contractible if the identity map Idx is homotopic to the constant map at some point of X. Prove the following. (a) X is contractible if and only if it is homotopy equivalent to a space consisting of a single point. (b) If X is contractible then it is simply connected¹. (c) If f and g are maps from Y into a contractible space X, then f~g. (ii) Recall C(X), the cone on X, is the space obtained from X × I by identifying X × {0} to a point. Prove that C(X) is contractible.