Illustrating Disconnection of Sets in R2

Illustrate the disconnection of two sets in bb{R}^2 using open sets to separate them.

In real analysis, the concept of disconnection is deeply tied to understanding how sets can be separated through the use of open sets. Two sets are disconnected if there exist open sets that contain each set, yet the intersection of these two open sets is empty. This problem is centered around illustrating this concept in the context of two-dimensional real space, $R^2$.

To approach this problem, it's important to think about the properties of open sets in $R^2$. Open sets can be envisioned as regions that don't include their boundary. For example, in $R^2$, an open ball or an open rectangle could serve as examples of open sets. The goal here is to find open sets that contain each of the given sets independently with no overlapping parts, illustrating their separation.

A common strategy is to consider the nature of the given sets and envision how open sets can contain them while ensuring that their intersection is empty. Visualizing the sets on the Cartesian plane can help in understanding spatial relations and imagining suitable open sets that satisfy the conditions for disconnection. This problem is fundamental for understanding more complex topological ideas where connectivity properties are key.