Limit Points in a Topological Space

Find the limit points and derive the set of a set $A$ in the topology $\tau$ on a set $X$, where $X = \{ A, B, C, D, E \}$.

This problem introduces you to the concept of limit points within a topological space. Understanding limit points is fundamental in topology as they help characterize the closure of a set. When attempting to determine the limit points of a set, it is crucial to consider the open sets that define the topology, as these open sets dictate the neighborhood structure around each point in the space. One effective strategy is to systematically examine whether each point in the space can serve as a limit point by checking if every open set containing the point also contains other points of the set. Analyzing limit points also offers insights into the closure of the set, which consists of all the limit points along with the original set itself. Exploring these concepts in the context of a finite set, as in this problem, provides a manageable entry point into more complex topological discussions you will encounter in further study.