# Category:Open Balls

This category contains results about open $\epsilon$-balls in the context of Metric Spaces.

Definitions specific to this category can be found in Definitions/Open Balls.

Let $M = \left({A, d}\right)$ be a metric space or pseudometric space.

Let $a \in A$.

Let $\epsilon \in \R_{>0}$ be a strictly positive real number.

The **open $\epsilon$-ball of $a$ in $M$** is defined as:

- $B_\epsilon \left({a}\right) := \left\{{x \in A: d \left({x, a}\right) < \epsilon}\right\}$

If it is necessary to show the metric or pseudometric itself, then the notation $B_\epsilon \left({a; d}\right)$ can be used.

## Pages in category "Open Balls"

The following 22 pages are in this category, out of 22 total.

### M

### O

- Open Ball in Cartesian Product under Chebyshev Distance
- Open Ball in Euclidean 3-Space is Interior of Sphere
- Open Ball in Euclidean Plane is Interior of Circle
- Open Ball in Euclidean Plus Metric is Subset of Equivalent Ball in Euclidean Metric
- Open Ball in Real Number Line is Open Interval
- Open Ball in Real Number Plane under Chebyshev Distance
- Open Ball in Standard Discrete Metric Space
- Open Ball is Convex Set
- Open Ball is Neighborhood of all Points Inside
- Open Ball is Open Set
- Open Ball is Subset of Open Ball
- Open Ball of Point Inside Open Ball
- Open Balls form Basis for Open Sets of Metric Space
- Open Real Interval is Open Ball
- Open Set may not be Open Ball