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**1. Homework Statement**

In topology, a f: X -> Y is continuous when

U is open in Y implies that f^{-1}(U) is open in X

Doesn't that mean that a continuous function must be surjective i.e. it must span all of Y since every point y in Y is in an open set and that open set must have a pre-image open in X, so there must be some x in X s.t. f(x) = y?

What is wrong with my logic?

**2. Homework Equations**

**3. The Attempt at a Solution**