In this example, by drawing a picture you can easily guess what $\Delta(p)$ is. You can then verify your guess formally using the method in the previous question.
Here's the geometric picture to draw. The set $Y$ is just the union of integer coordinate lines in $\mathbb{R}^2$: horizontal lines of the form $\mathbb{R} \times m$, $m \in \mathbb{Z}$; and vertical lines of the form $n \times \mathbb{R}$, $n \in \mathbb{Z}$. The integer lattice points $\mathbb{Z}^2$ divide up the horizontal lines into segments $[n,n+1] \times m$, and they divide up the vertical lines into segments $n \times [m,m+1]$ (for all $(m,n) \in \mathbb{Z}^2$).
From the formula $p(x,y) = e^{2 \pi i y} - e^{2 \pi i x}$, you can see that every horizontal segment $[n,n+1] \times m$ maps once around the $v$ loop of $X$, and furthermore as a $(x,m)$ moves to the right, $x \in [n,n+1]$, the point $p(x,m)$ moves in the counterclockwise direction around the $v$ loop. Similarly, every vertical segment $n \times [m,m+1]$ maps once around the $u$ loop, and as $(n,y)$ move upward the point $p(n,y)$ moves in the counterclockwise direction around the $u$ loop. To indicte this, make each horizontal segment into a rightward pointing arrow labelled by the letter $v$, and make each vertical segment into an upward pointing arrow labelled by the letter $u$.
Now, stand back and gaze at your beautiful picture.
Look for all the symmetries of the picture.
You should be able to see that any translation of the plane with an integer displacement vector $(i,j) \in \mathbb{Z}^2$, displacing $i$ horizontal units and $j$ vertical units, is a deck transformation, taking each horizontal segment $[n,n+1] \times m$ with its rightward pointing $u$ arrow to another horizontal segment $[n+i,n+1+i] \times (m+j)$ with its rightward pointing $u$ arrow, and similarly for the vertical segments with their upward pointing $v$ arrows.
