The author of the question being raised in Proving that $C$ is a subset of $f^{-1}[f(C)]$ received some similar proofs. I can follow them but there's something I don't understand.
It's probably a stupid question, but could someone explain me why $f(x)$ should be defined for all $x$? Like when we have $F=[(0,1)]$ on $X=[0,2]$ then $F^{-1}(F(\{0,2\}))=F^{-1}(\{1\})=\{0\}$. Where is my mistake?
The function $f$ is defined where ever it's defined. However, talking about the image of a set $S$ under $f$ only makes sense if $f$ is defined on $S$. If you restrict $f$ to $S$ you have equality: $$S=f^{-1}(f(S)).$$ Now, if you look at $f:T\longrightarrow M$, where $M$ and $T$ are sets and $S\subset T$, you might have some $s\in f(S)$ with $f(t)=s$ where $t\in T\setminus S$ and thus $t\in f^{-1}(f(S))$, but $t\not\in S$, so you get inclusion: $$S\subseteq f^{-1}(f(S)).$$ Does this help?