A confusion regarding the definition of continuous functions.

Question

Let us suppose $f:X→Y$ is a continuous, non-surjective mapping.

Also assume that there is an open set $A\subset Y$ which contains points which are not in $f(X)$. What would $f^{−1}(A)$ be? Would it even be defined?

Although this might seem trivial, reading an answer somewhere has caused me to question such basics.

Thanks in advance!

Answer 1

Suppose that $A$ is such that $f(x)\notin A$, for all $x\in X$. Then we have: $$f^{-1}(A)=\{x\in X\mid f(x)\in A\}=\varnothing.$$

There is nothing wrong with sets having an empty preimage.

Answer 2

You just consider $f^{-1}(A) \cap X$. You can consider it as $f^{-1}(A\cap Y)= f^{-1}(A)\cap X$ , since $f^{-1}(Y)=X$