How to find the preimage of a set I'm trying to prove continuity of two different functions $f\colon A\to B$. I know that to find continuity, the inverse of all open subsets in $B$ must be open in $A$. I also know that to find the inverse of an open subset you have to find the preimage. I just don't know how to find the preimage. How do you find the preimage? 
For example, if the set is the discrete topology of $\{1,2,3\}$, how would you find the preimage of $\{1\}$, $\{2\}$, etc. The same for the indiscrete topology. How would you find the preimage of the null set and the whole set? Thank you!

Let $f:A\to B$ be a map and let $S$ be a subset of $B$, then the preimage of $S$ under $f$ is
$$f^{-1}(S)=\{a\,|\,f(a)\in S\}.$$
For example if $A=\{1,2,3\}$, $B=\{a,b,c\}$ and $f$ is map defined by $f(1)=f(2)=a$, $f(3)=b$, then 
$$f^{-1}(\emptyset)=\emptyset,\\
f^{-1}(\{a\})=\{1,2\},\ 
f^{-1}(\{b\})=\{3\},\ 
f^{-1}(\{c\})=\emptyset,\\
f^{-1}(\{a,b\})=\{1,2,3\},\ 
f^{-1}(\{a,c\})=\{1,2\},\ 
f^{-1}(\{b,c\})=\{3\},\\
f^{-1}(\{a,b,c\})=\{1,2,3\}.$$
