I'm starting to learn about etale groupoids and I have this specific question about topological groupoids. It should be quite simple but I'm having trouble showing it. Note $G_0$ is denoting the units in $G$.
I start with a point $x \in (U \cap G_0)G$. So $x = yz$ with $y = 1_a$ the identity of some point in $G_0$ and also inside of $U$. Since we are assuming that the composition of $y$ and $z$ is possilbe it is necessary $d(z) = r(y) = a$. Moreover since $y$ is a unit then $x = z$.
Now what I want to do is construct some open sets $X$ and $Y$ that we have on the right. But since the topology of $G$ is arbitrary I'm not sure how we can do this. I was thinking we can take an open neighbourhood of $x$ say $V$. Then we know that $1_a \in VV^{-1}$ and $1_a \in U$. However what about everything else in $V$, how could we show $VV^{-1} \in U$.