# Category:Open Balls

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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 $M = \struct {A, d}$ 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:

- $\map {B_\epsilon} a := \set {x \in A: \map d {x, a} < \epsilon}$

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

## Subcategories

This category has the following 6 subcategories, out of 6 total.

## Pages in category "Open Balls"

The following 44 pages are in this category, out of 44 total.

### C

### E

### M

### O

- Open Ball Centred at Origin in Normed Vector Space is Symmetric
- Open Ball Contains Open Ball Less Than Half Its Radius
- Open Ball contains Smaller Open Ball
- Open Ball contains Strictly Smaller Closed Ball
- 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 Infinite-Dimensional Normed Vector Space is not Weakly Open
- Open Ball in Normed Division Ring is Open Ball in Induced 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 Connected
- Open Ball is Convex Set
- Open Ball is Neighborhood of all Points Inside
- Open Ball is Open Set
- Open Ball is Path-Connected
- Open Ball is Simply Connected
- Open Ball is Subset of Open Ball
- Open Ball of Point Inside Open Ball/Metric Space
- Open Ball of Point Inside Open Ball/Normed Vector Space
- Open Ball of Point Inside Open Ball/Pseudometric Space
- Open Balls form Basis for Open Sets of Metric Space
- Open Balls form Local Basis for Point of Metric Space
- Open Balls of Same Radius form Open Cover
- Open Balls of Supremum Metric on Continuous Real Functions on Closed Interval
- Open Balls whose Distance between Centers is Twice Radius are Disjoint
- Open Real Interval is Open Ball
- Open Set may not be Open Ball