The formal process behind « cutting » surfaces in topology

I am an undergraduate student in mathematics and decided to focus on topology, algebraic topology, and I decided to start with surfaces. I am reading a few books about the subject and a lot of proofs use arguments that sound like : « by cutting [a surface] along [a simple curve], we get something homeomorphic to [some known thing] which gives us ... ». This arguments are generally convincing visually but I couldn’t manage to fill in all the formal details of such proofs if I were asked to. I understand better the process of pasting, which I formalize as a disjoint union and a quotient, but I am not sure how to formalize this « cutting » process, as intuitive as it is. If one wants to know, the proofs I read that contain this type of reasonning are in « A Primer on Mapping Class Groups » , Farb & Margalit.

Thanks a lot !

• Maybe the paragraph You May Have Been Provoked to Perform an Illegal Operation (page 122) from Viro-Ivanov-Netsvetaev-Kharlamov book will be useful here. They try to go into full details about what gluing two topological spaces means. – Evgeny Mar 20 '18 at 13:44

Cutting is the reverse process: if you can cut $C$ along a curve to get $A$ and $B$, it follows that by gluing together $A$ and $B$ and quotienting out the seam along $A$ and $B$ where the cut occurred, you get $C$ again.
So, to prove that cutting $C$ gives $A$ and $B$, it's enough to use your knowledge of gluing to show that $A$ and $B$ can be joined together and quotiented at the seam to produce $C$.