Education: Metric space: open Ball (sphere or disk) & its examples.

16/10/2014

Metric space: open Ball (sphere or disk) & its examples.

Metric space:

Definition: Open Ball

Motivation

The open ball is the building block of metric space topology. We shall define intuitive topological definitions through it (that will later be converted to the real topological definition), and convert (again, intuitively) calculus definitions of properties (like convergence and continuity) to their topological definition. We shall try to show how many of the definitions of metric spaces can be written also in the "language of open balls". Then we can instantly transform the definitions to topological definitions.

Definition

Given a metric space

(X, d)

an open ball with radius

r

around

p

is defined as the set

B_r(p) = \{ x \in X \mid d(x,p) < r \}, (r \in \mathbb{R}^{+})

.

Intuitively it is all the points in the space, that are less than

r

distance from a certain point

p

.

Examples

Why is this called a ball? Let's look at the case of

\mathbb{R}^3

:

d ((x_1,x_2,x_3),(y_1,y_2,y_3)) = \sqrt{(x_1-y_1)^2 + (x_2-y_2)^2 + (x_3-y_3)^2}

.

Therefore

B_r ((0,0,0))

is exactly

{x_1}^2 + {x_2}^2 + {x_3}^2 < r^2

- The ball with

(0,0,0)

at center, of radius

r

. In

R^3

the ball is called open, because it does not contain the sphere (

{x_1}^2 + {x_2}^2 + {x_3}^2 = r^2

).

The Unit ball is a ball of radius 1. Lets view some examples of the

B_1((0, 0))

unit ball of

\mathbb{R}^2

with different p-norm induced metrics. The unit ball of

\mathbb{R}^2

with the norm

||\cdot ||_p

is:

B_1((0, 0)) = \{ (x,y) \in \mathbb{R}^2 \mid d((x,y),(0,0)) < 1 \}

=

\{(x,y) \mid ||(x,y)-(0,0)||_p < 1 \} = \{(x,y) \mid ||(x,y)||_p < 1 \}

=

\{(x,y) \mid \sqrt[p]{|x|^p + |y|^p} < 1\}

The metric induced by $||\cdot ||_1$ in that case, the unit ball is: $|x| + |y| < 1$

The metric induced by $||\cdot ||_2$ in that case, the unit ball is: $\sqrt{x^2 + y^2} < 1$

The metric induced by $||\cdot ||_\infty$ in that case, the unit ball is: $\max\{|x| , |y|\} < 1$

As we have just seen, the unit ball does not have to look like a real ball. In fact sometimes the unit ball can be one dot:

The discrete metric, The unit ball is $B_1((0, 0)) = \{d((0,0),(x,y)) < 1\} = \{d((0,0),(x,y)) = 0 \} = \{(0,0)\}$