 Solution

ID: past-exam-of-the-topology-course-of-the-university-of-oxford/2015/1/a/iii/solution

f (A) = U \cup V

A \cap f^{- 1} (U), A \cap f^{- 1} (V)

would be disjoint nonempty relatively open sets covering

A

. They are relatively open by continuity of the restriction

f ∣_{A} : A \to f (A)

. This contradicts connectedness of

A

A connected, f continuous ⟹ f (A) connected .

This is the continuous image of a connected space theorem; no separation property of the target is required.

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