A closed ball in a metric space is the set of all points whose distance from a given center point is less than or equal to a specified radius. This concept is crucial in understanding open and closed sets, as a closed ball contains its boundary, differentiating it from an open ball, which does not. Closed balls play a significant role in topology by helping define closed sets and providing insight into the structure of metric spaces.
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In a two-dimensional Euclidean space, a closed ball centered at point (a, b) with radius r includes all points (x, y) such that the distance $$ ext{d}((x,y), (a,b)) \leq r$$.
Closed balls are always closed sets in a metric space, meaning they contain all their limit points.
The concept of closed balls extends to higher dimensions, where it can be represented as all points within a certain distance from a center in n-dimensional space.
In a discrete metric space, every closed ball contains only the center point when the radius is less than 1.
Closed balls can be used to demonstrate various properties of continuity and compactness in topology.
Review Questions
How does a closed ball differ from an open ball in the context of their boundaries?
A closed ball includes all points within a specified distance from its center, including the boundary points that are exactly at that distance. In contrast, an open ball consists only of those points strictly less than that distance, thus excluding the boundary. This distinction highlights the difference between closed sets, which contain their boundaries, and open sets, which do not.
Discuss how closed balls relate to the concept of closed sets within metric spaces.
Closed balls are foundational examples of closed sets in metric spaces because they include their boundary points. By definition, a set is considered closed if it contains all its limit points. Since closed balls inherently satisfy this condition, they provide critical insight into identifying and constructing other closed sets within a given metric space.
Evaluate the significance of closed balls in understanding continuity and compactness in topology.
Closed balls play a vital role in analyzing continuity and compactness due to their structural properties. For example, if a function is continuous on a closed ball, it ensures that the image of that closed ball under the function will also be compact. This relationship illustrates how closed balls facilitate discussions about convergence and limits, which are central themes in topology. Thus, recognizing how closed balls function within these contexts allows for deeper comprehension of topological properties.
Related terms
Open Ball: An open ball is the set of points in a metric space that are within a certain distance from a center point, but does not include the boundary points.
A metric space is a set equipped with a metric, which is a function that defines a distance between any two points in the set.
Boundary: The boundary of a set consists of points that can be approached from both inside and outside the set, and it plays a key role in differentiating between open and closed sets.