Suppose we have the pair groupoid $G:\mathbb{R}^2\rightrightarrows \mathbb{R}$ which is a Lie groupoid with source $s$ and target $t$ maps given by the first and second projection, respectively. Recall that a local bisection $\sigma=(s,U)$ of $G$ is a section $\gamma: U\subset \mathbb{R}\to \mathbb{R}^2$ of $\pi_1$ such that $\pi_2\circ \gamma: U\to U$ is a diffeomorphism and $U$ is an open subset of $\mathbb{R}$. Define the following relation in the set of local sections of $G$. $(\gamma,\delta)\in R$ if and only if there is a $y\in dom(\delta)$ such that $\gamma(y)=\delta(y)$. Clearly $R$ is not transitive. So, consider its transitive closure $R^+$. My question is:
Is $\bigcup_{(\gamma,\delta)\in R^+} \gamma(dom(\delta))$ an open set of $\Omega(\mathbb{R}^2)$ (the usual topology of $\mathbb{R}^2$)? My good feeling says yes, but don't find a good proof to explain this yet. Helps are appreciated.
