Quotient maps, open and closed maps

[runnnincircles]

I am confused about the definition of a quotient map. If X and Y are topological spaces, then p:X-->Y is a quotient map if it is surjective and continuous and satisfies the following condition: A subset U of Y is open/closed in Y iff p-1(U) is open/closed in X

Comments

[phlebas]

Your thinking in the p.s. is correct - can you come up with a counterexample?

[runnnincircles]

I think i understand my mistake. I assumed (without good reason) that p was injective.