Adjunction space and quotient topology

Let X and Y be disjoint spaces, and A⊆X a subset.
Let f: A→Y be a function. Let Z=(X-A)∪Y be the disjoint union of sets. Then there is a natural surjection from X∪Y(as topological sum) to Z. The quotient topology on Z induced by this function is none other than the adjuction space X∪_fY.