Does $X \lor Y = (X \sqcup Y)/\{x_0, y_0\}$ depends on the points $x_0, y_0$?

Question

Let $X, Y$ two topological spaces and $x_0,y_0$ in $X, Y$ respectively. We denote the wedge product of $(X, x_0)$ and $(Y, y_0)$ by $X \lor Y$, such that $$X \lor Y = (X \sqcup Y)/\{x_0, y_0\}.$$ This might by a silly question, but why there is no mention of $x_0, y_0$ in the notation of the wedge product $X \lor Y$? Is it because it does not depend on $x_0, y_0$ or just because mathematicians are lazy?