I'm reading over the following proof as seen here.
My possibly silly question is, what is meant by $\delta R_{1}$ here?
I'm trying to understand the reasoning behind the proof, but am getting stuck by this notation I haven't encountered... $x$ does not belong to $R_{1}$... is it within some $\delta$ of $R_{1}$? If so, how can we select any $\epsilon > 0$?
On
It's the boundary of $R_1$. Although I have never seen the notation $\delta R_1$ used for boundary (I would prefer $\partial R_1$ or just $\text{bd}(R_1)$), I can be sure because the next phrase in the sentence recalls the definition of a boundary: neighborhoods of its points intersect the set and its complement. Anyway they're both forms of delta so it's not too outlandish a notation.
I believe $\delta R_1$ refers to the boundary of $R_1$ here. The notation $\partial X$ for the boundary of $X$ is a bit more common.