What is the relationship between topological gluing and quotient space?

Question

From what I understand, you take 2 subspaces of different topological that have similar open sets in a sense. That is, if you link up the elements, the open sets will also link up. Is this correct? Where does the quotient space come in here?

Answer 1

The gluing space is a particular case of quotient space. Take two topological spaces $X,Y$ and a subspace $A\subseteq X$ with a continuous map $f:A\to Y$. In orther to construct the gluing space associated to $f$, you "glue" $A$ and $f(A)\subseteq Y$.

The way to see this as a quotient space is taking the disjoint union $X\sqcup Y$ and the equivalence relation $\{A\sim f(A)\}$. Thus, the gluing space is the quotient $(X\sqcup Y)/\{A\sim f(A)\}$.