Understanding Isolated Points
In topology, the concept of isolated points is crucial for understanding the structure of sets in a given space. Let's explore this in detail.
Definition of Isolated Points
- An isolated point of a set is a point that is not a limit point of the set.
- In simpler terms, a point is isolated if there exists a neighborhood around it that contains no other points from the set.
Characteristics of Finite Sets
- A finite set contains a limited number of elements.
- Each element in a finite set can be considered isolated because you can always find a small enough neighborhood around each point that includes no other points from the set.
Comparison with Other Point Types
- Limit Point: A point is a limit point if every neighborhood around it contains at least one point from the set different from itself. This is not applicable to finite sets, as they do not have other points infinitely close.
- Interior Point: An interior point is one where there exists a neighborhood entirely contained within the set. In a finite set, no point can be an interior point because you cannot find such a neighborhood without including points outside the set.
- Exterior Point: An exterior point is a point that is not in the set and does not belong to any neighborhood of the set. This concept is not directly related to the points within a finite set.
Conclusion
Thus, every point in a finite set is indeed an isolated point, making option 'C' the correct answer. Each point stands alone without being close to any other point of the set, reinforcing the definition of isolation in the context of finite sets.