Disconnected Sets in Real Numbers

Demonstrate that a set in $\mathbb{R}$ is disconnected if open sets $U_1$ and $U_2$ can be found such that the intersection of $e$ with both $U_1$ and $U_2$ are non-empty, and their union covers all of $e$, while the intersection of these parts is empty.

In this problem, you're tackling the concept of disconnected sets in the context of real analysis, specifically within the realm of topology on the real line. A set is disconnected if it can be divided into two non-empty open sets such that their union reconstructs the original set, but their intersection is empty. This idea is essential in understanding how sets can be partitioned in a way that they are 'separated' from each other.

To demonstrate that a set is disconnected, you'll typically want to identify two open subsets within your given set that satisfy the conditions mentioned. These subsets should share no points (i.e., their intersection is empty), yet their combination should encompass the entire original set. This exercise enriches your comprehension of how topology allows the classification of sets not by size or distance, but by their properties of separation.

Exploring disconnected sets provides insight into broader discussions about connectedness, which is a cornerstone in the study of continuous functions and topological spaces. When a set is connected, it cannot be split into two such non-empty, disjoint open sets. Hence understanding disconnectedness is pivotal in recognizing and proving the connectedness (or lack thereof) of different spaces in real analysis.