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For , every Euclidean closed ball and every Euclidean sphere of positive radius is compact
Statement
For , , and , the Euclidean closed ball and Euclidean sphere are compact.
Facts & Assumptions
Given: , , and .
The sets and are respectively the points satisfying and (Euclidean spheres and closed balls as subspaces of ).
The Euclidean norm is continuous and satisfies (The finite and reverse triangle inequalities for a norm; and for every norm on satisfies and is Lipschitz, hence continuous, for ).
The Euclidean compactness theorem identifies compactness with closedness and boundedness (For a nonempty subset of with , compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent).
Euclidean open sets are the metric-open sets, and metric boundedness means containment in a ball (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space).
Proof
Both sets are bounded: lies in the ball of radius about , and .
The complement of is open: if , then the ball about of radius stays in the complement by [L2].
The complement of is open: if , then a ball about of radius avoids the sphere by [L2].
Thus both sets are closed and bounded, hence compact by [L3].
Depends on
- Euclidean spheres and closed balls as subspaces of $\mathbb{R}^n$
- For a nonempty subset of $\mathbb{R}^n$ with $n\ge1$, compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent
- The finite and reverse triangle inequalities for a norm; and for $n \ge 1$ every norm $N$ on $\mathbb{R}^n$ satisfies $N(x) \le C\lVert x\rVert_1$ and is Lipschitz, hence continuous, for $d_2$
- The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement
- Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space
Used by
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 147 results over 25 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Heine-Borel theorem (standard reference, not scraped)