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Prove that if f: X → Y is a continuous function and C ⊂ Y is...

Prove that if f: X → Y is a continuous function and C ⊂ Y is closed that the preimage of C, f^-1(C), is closed in X.

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Answer #1

Given be a continuous function .

Suppose is closed . We need to prove that is closed in X .

Let   be a sequence in f-1(C) converges to x

The sequence (f(xn)) converges to f(x) since f is continuous so image of a convergent sequence is convergent .

Since the sequence (f(xn)) belongs to C and C is closed so ,

f(x) C

x f-1(C)

Since the sequence (xn) is arbitrary so for every sequence (xn) in f-1(C) converges to x then x f-1(C) .

So f-(C) contains all its limit points and hence is closed if C is closed .

.

.

If you have any doubt or need more clarification at any step please comment .

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