Final answer:
Isolated points are always boundary points because by definition, they have neighborhoods containing no other points of the set they belong to, which also means these neighborhoods include points not in the set, meeting the criteria of a boundary point.
Step-by-step explanation:
In understanding the relationship between isolated points and boundary points, it is essential to recognize the definitions and implications of these concepts in topology. An isolated point of a set S in a topological space is one that has a neighborhood that does not contain any other points of S. On the other hand, a boundary point of S is a point where every neighborhood of it contains at least one point in S and at least one point not in S.
Let’s consider a set S and a point x. If x is an isolated point of S, by definition, we can find a neighborhood around x that contains no other points of S. However, this neighbourhood will contain points not in S, implying that x is also a boundary point. This is because boundary points are characterized by the fact that their neighborhoods always include points in the set and points external to it. Therefore, when x is isolated, it inherently fulfills the criteria of being a boundary point, as every neighborhood around x contains points that are not in S.
To solidify this concept with an example, imagine a set S consisting of discrete points on the real number line, such as the set {1, 2, 3}. The point 2 is an isolated point since there is a segment around it (e.g., (1.5,2.5)) that contains no other points of S. The same segment will inevitably contain points not in S, like 1.6 or 2.4, thus making 2 a boundary point as well. This illustrates that while boundary points can represent a larger set of conditions, every isolated point will always satisfy the definition of a boundary point.