# Continuous functions in topology

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

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Hurkyl
Staff Emeritus
Gold Member
If you think it isn't true, the try and construct a counterexample.

matt grime
Homework Helper
So you're asking about the case when there is no x in X st... That would seem to be the empty set. Now what was the definition of topology?

I think matt grime answered it, but my counter example is:

f:R->R
f(x) = abs(x)

the set (-4, -2) is open but when when you pull it back you get the null set, which is open in the usual topology on R. I think I see now!