Yes, a closed disk is compact.
In the realm of topology and mathematical analysis, the concept of compactness is fundamental. A set is considered compact if, intuitively, it is both "closed" and "bounded." This property ensures that the set behaves nicely under various mathematical operations, particularly concerning sequences and continuous functions.
Understanding Compactness
A topological space (or a subset of one, like a disk in Euclidean space) is compact if every open cover of the space has a finite subcover. While this definition is standard, a more intuitive understanding, especially in Euclidean spaces ($\mathbb{R}^n$), relates to the concepts of being closed and bounded.
- Closed: A set is closed if it contains all its limit points. For a disk, this means including its boundary (the circle itself).
- Bounded: A set is bounded if it can be entirely contained within some finite "ball" or region. A disk, by its nature, has a finite radius, making it bounded.
Why a Closed Disk is Compact
The compactness of a closed disk stems directly from these two properties.
The Intuitive Explanation (from Reference)
As stated in the reference, in two dimensions (like a standard disk on a plane):
"...closed disks are compact since for any infinite number of points sampled from a disk, some subset of those points must get arbitrarily close either to a point within the disc, or to a point on the boundary."
This statement highlights a crucial aspect of compactness known as sequential compactness. It implies that any infinite sequence of points within a closed disk will always have a convergent subsequence whose limit point is also within the disk (or on its boundary). This prevents "points from escaping" or "spreading out infinitely" within the set.
The Formal Definition: Heine-Borel Theorem
For subsets of Euclidean space ($\mathbb{R}^n$), the intuitive understanding aligns perfectly with a powerful result: the Heine-Borel Theorem. This theorem states that a subset of $\mathbb{R}^n$ is compact if and only if it is both closed and bounded.
Since a closed disk (which includes its boundary) has a finite radius and is therefore bounded, and it includes all its limit points (itself and its boundary), it satisfies both conditions.
Closed vs. Open Disks: A Comparison
The distinction between a closed and an open disk is crucial for understanding compactness.
| Feature | Closed Disk | Open Disk |
|---|---|---|
| Definition | All points strictly inside plus all points on the boundary. | All points strictly inside (boundary excluded). |
| Notation | ${ (x,y) \mid x^2+y^2 \le r^2 }$ | ${ (x,y) \mid x^2+y^2 < r^2 }$ |
| Closed? | Yes | No (does not contain its boundary limit points) |
| Bounded? | Yes | Yes |
| Compact? | Yes (Closed + Bounded) | No (Bounded but not Closed) |
| Example | A coin, including its edge. | An open circle drawn on paper, excluding the line. |
An open disk, while bounded, is not compact because it is not closed. Points can approach the boundary from within the disk, but the boundary points themselves are not included in the set, meaning sequences can converge to points outside the set.
Practical Significance
The property of compactness is incredibly important in various areas of mathematics, including:
- Analysis: It guarantees the existence of maximum and minimum values for continuous functions defined on compact sets (Extreme Value Theorem).
- Optimization: Many optimization problems involve finding extreme values, and compactness of the feasible region simplifies the analysis.
- Topology and Geometry: It plays a role in defining many important concepts and proofs.
In summary, a closed disk is a classic example of a compact set in Euclidean space, demonstrating the fundamental topological properties of being both closed and bounded.
