Is $f^{-1}(A)$ always rigorously defined even when $A$ is a set not in the codomain

Question

Let $f:X\to Y$ be a function. We know that $f^{-1}(y)$ is not always defined.

The codomain is $Y$.

Can I say that $f^{-1}(A)$ is always well-defined unconditionally for an arbitrary set $A$?

Do we need any topological assumptions?

For example, if $A$ does not intersect with the codomain, I can simply say that $f^{-1}(A)=\emptyset$. But when I check the textbook definition for "inverse image function" they always require $A\subseteq Y$. I need reference for the case when $A$ is not in $Y$.

Answer 1

I think you have fallen victim to notational abuse, unfortunately.

A function $f\colon X \to Y$ always determines two functions:

1. The "direct image function" $\mathsf{f}\colon P(X)\to P(Y)$ (where $P(S)$ is the power set of $S$, the set of all subsets of $S$), which is defined as follows: given a subset $A\subseteq X$, we have $$\mathsf{f}(A)=\{y\in Y\mid \text{there exists }a\in A\text{ such that }f(a)=y\}.$$
2. The "inverse image function", $\mathscr{F}\colon P(Y)\to P(X)$, defined as follows: given a subset $B\subseteq Y$, we let $$\mathscr{F}(B)=\{x\in X\mid \text{there exists }b\in B\text{ such that }f(x)=b\}.$$

By abuse of notation, we often denote the direct image function $\mathsf{f}$ with the same symbol as the original function (here $f$), and write $$f(A) = \{f(a)\mid a\in A\}.$$ And also by abuse of notation, we often denote the inverse image function $\mathscr{F}$ by "$f^{-1}$" (even though this should be restricted to the inverse function and only used when $f$ has an inverse, i.e. is bijective), and write $$f^{-1}(B)=\{x\in X\mid f(x)\in B\}.$$

To compound the abuse even more, when $A=\{a\}$ is a singleton, we write $f(a)$ instead of $f(\{a\})$, and rely on context to tell us if we mean the value of $f$ at $a$ or the direct image of the singleton. Likewise (and also in an exhibition of abuse upon abuse) if $B=\{b\}$ is a singleton, we write $f^{-1}(b)$ instead of $f^{-1}(\{b\})$ instead of $\mathscr{F}(\{b\})$ . To make matters even worse, sometimes when $f$ is injective but not surjective, we also use $f^{-1}$ to denote the inverse of the related (but under some formalisms, technically different from $f$) function $f\colon X\to f(X)$.

It is the latter abuse of abuse of abuse of notation which makes you say that sometimes "$f^{-1}(y)$ is not defined", and leads some to reply "no, it is defined, it is all elements that map to $y$, and could be empty."

So, the inverse image function is defined at every subset of $Y$ and gives a (possibly empty) subset of $X$. The inverse function may or may not exist, or be defined at specific elements.

This abuse can be confusing and unfortunate when you are just starting. It is a good idea, at that time, to be very careful to make sure there is no confusion. As you gain experince and comfort, the notational abuse is rather convenient and not confusing, which is why it is so prevalent.

But in no notation, abusive or otherwise, does it make sense to talk about $f^{-1}(Z)$ if $Z$ is not a subset (or an element) of $Y$. It's like asking for the floor of $1+i$, $\lfloor 1+i\rfloor$, a nonsensical question. You could, as Severin Schraven suggest in comments, further abuse notation even more (is there any kind of shelter for abused notations to escape to where they can be safe?) and simply think of $f^{-1}(Z)$ as "the set of all $x\in X$ such that $f(x)\in Z$, so that elements of $Z$ that are not in $Y$ will be irrelevant, but still write it and say it is "defined" (rather abusively...)