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Is there a map $A\rightarrow B$ that is injective but not surjective?

A friend asked me about the following statement:
There is no map $f:A\rightarrow B$ that is surjective but not injective.
This is what I think but I am not sure if it is correct:
Let us assume the statement to be false then it is possible to find a map $f:A\rightarrow B$ which is injective but not surjective. Assume that $x\in A$ and $f(x)=0$, then $f$ is injective, but $f$ is not surjective.
Can someone please explain how to show that the statement is false? Is it easy to do or is it a trick that is hard to grasp?

A:

Suppose there is such a map $f : A \to B$. Then it is injective, and therefore the image $f(A) = f \left( \bigcup_{a \in A} \{a\} \right)$ is the union of a disjoint collection of open sets in $B$. Since $B$ is compact, its open sets cover $B$, and so there is some subset of $A$ that meets every open set of $B$. Since $f$ is not surject

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