Question: please answer #1 Notation and basic conventions. Throughout let X and Y denote topological spaces, and for a subset A of a topological space denote

please answer #1

Notation and basic conventions. Throughout let X and Y denote topological spaces, and for a subset A of a topological space denote its closure by A and the set of its limit points by A'. 1. a. Prove that a topological space X is Hausdorff if and only if the set A - {(z, r); I E X} is a closed subset of X x X. (One direction of "if and only if" suffices.) b. Consider the statement: If f: XY and for every A C X, f(A) C f(A), then f is continuous. (i) State the contrapositive of this statement. (ii) Prove this statement (or its contrapositive)

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