Question

Problem II. a) Show that f: X →Y is continuous if and only if f-'(C) CX is closed for every closed C CY b) Then show that a function f: X + Y is continuous if and only if f(A) < f(A) for all ACX

For metric spaces and topology

Answer 1

Let f: x y be a continuous function be Let Then Y-C is closed set. any open set inr So f'(Y-S) is open in x Now $(Y-C) = f (Y) - F-1(C) x-f-(0) X-f-(e) is open in x => f-(C) is closed :: for all closed cey, f-(C) is closed in X. convensely het f'(() is closed in a for 011 Closed C SY, set, so in X. 9 Let ucy be any open Y-u is closed in Y. by assumption f-'(4-0) is closed in a - f-(Y)-f(0) is closed in x f-'(U) open in Hence f '(U) is open is closed in a -> f-'(U) is x for all set ucx open so f is continuous function.

Let A cx be NO 50 let f: x y be continuas function, ang subset. .. f(A) CY f(A) < FCA → f"(f(as) st"(FC) again Asf(f(A)) A s f'(fcar) = f'(F(a) → As f(TA) Since f(A) is closed iny and f is continuous , then f-'( F(A)) closed Again that A is the smallest closed set containing A and f'(FTA)) is a containing Hence Ā = f'( FCA) Ź f(A) < FCA) por exsolya Let f(A) S f(a) for all A CX, in & know closed set set in Y. Let k be a closed Now f-'(K) SX generally $""(x) Ş f*(K) By assumption , f(FM) = f(f(a) .: f'() SX

Since *5 (C*).+) → fcf'(x)) sū f(f"(k)) sk [*=*.:] (3) ск From @ and f(F"(x)) = f(f(x)) = K = f(f='(W)) f"(K) = f'(K) f'(K) = f'(x) => f-(K) is closed in x 4 ® and 6) we have Thus f is a continuous function.

This is true in topological space and metric space is special case of topological space so these are true in metric space.

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