 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2009/e/e2/a/ii/solution

Let

(U_{i})_{i \in I}

be an open cover of

f (X)

, with the sets open in

Y

f^{- 1} (U_{i})

form an open cover of

X

, because

f

is continuous. By compactness, finitely many of these preimages cover

X

. Their corresponding sets

U_{i}

cover

f (X)

, providing a finite subcover. Hence the continuous image is compact.

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