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Understanding boundary points in mathematics is crucial for analyzing properties of sets in real analysis. Boundary points are the 'edge pieces' of a set, lying just at the interface where the set meets its complement.

For a set S, boundary points are defined as those points where every open neighborhood contains at least one point in S and at least one point not in S. In the textbook example (a), the set includes intervals and a single point, which results in boundary points exactly at the edges of these intervals and the single point itself, specifically \-2, 2, 3, 4, 7\. These boundary points act like a frame, marking the transitions from the set to what lies beyond.

Understanding and identifying the boundary points of a set like \(S\) provide foundational insights which are essential before moving on to complex topics such as closure and the nature of isolated points within the set. Recognizing 'edge' and 'external' elements lays a solid groundwork for further analysis in topology and real analysis.

Closure is a fundamental concept that solidifies our understanding of sets in real analysis. Intuitively, the closure of a set encompasses the set itself along with all the boundary points that 'hug' the set. It effectively 'seals' a set by including all limit points, indicating that you cannot 'escape' the set by infinitesimally small movements.

In the context of our textbook exercise, the closure of set \(S\) for example (a) is the interval and points covered by \(S\) plus all its boundary points, resulting in \((-\textbackslash\textbackslash infty,-2] \cup [2,4] \cup (7,\textbackslash\textbackslash infty)\). The idea of closure is particularly important because it connects to various other areas such as continuity, compactness, and convergence in real analysis. When we talk about closures, we are essentially dealing with the concept of 'completeness' of a set, which is pivotal in various theorems and proofs in advanced mathematics.

In the landscape of real analysis, isolated points are like solitary islands in the sea of numbers. An isolated point in a set S is one that stands alone, with no neighboring points from S in its immediate vicinity. Mathematically, this means for some distance around an isolated point, you can place an open interval that contains no other points of S.

In exercise example (a), the number 4 is such an isolated point. It's part of the set, yet it has a 'buffer zone' around it, with no other members of S encroaching its space. Identifying isolated points helps us understand the 'sparseness' or 'disconnectedness' of a set and has implications in continuity and differentiability within calculus. In example (b), each integer is an isolated point because you can always find an open neighborhood around any integer that doesn't contain any other integers from the set.

When studying isolated points, we gain insights into the granularity of a set and its underlying structure, which are critical for nuanced comprehension and application in mathematical proofs and real-world problem-solving.