Example · High school
Terms
| Term | This example | Why |
|---|
| interior |
$\operatorname{int} D = \setdef{(x, y) \suchthat x^2 + y^2 < 1}$
|
The interior of the closed unit disk $D$ is the open disk.
|
| boundary |
$\partial D = \setdef{(x, y) \suchthat x^2 + y^2 = 1}$
|
The boundary of the closed unit disk $D$ is the unit circle.
|
| exterior |
$\operatorname{ext} D = \setdef{(x, y) \suchthat x^2 + y^2 > 1}$
|
The boundary is the unit circle, which $D$ contains, and the exterior is everything outside the circle.
|
| area |
$\pi$
|
The area of the closed unit disk is $\pi r^2$ with $r = 1$.
|
| diameter |
$2$
|
The diameter of the closed unit disk is $2$, the distance between two opposite points of its boundary.
|
| cardinality |
$\mathfrak{c}$
|
The cardinality of the closed unit disk is $\mathfrak{c}$, the same as that of $\reals$.
|
| measure |
$\pi$
|
The two-dimensional Lebesgue measure of the closed unit disk is its area, $\pi$.
|
| Hausdorff dimension |
$2$
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The Hausdorff dimension of the closed unit disk is $2$, as for every region of the plane with nonempty interior.
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| compact |
✓ |
|
| closed ball |
✓ |
|
| connected |
✓ |
|
| path connected |
✓ |
The closed unit disk is path-connected, and so connected, because the straight segment between two of its points stays in the disk.
|
| simply connected |
✓ |
The closed unit disk is simply connected because every loop in it can be shrunk to a point.
|
| convex set |
✓ |
The closed unit disk is convex, because the segment between two points of the disk stays in the disk.
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