# Definition:Open Set/Metric Space

## Definition

Let $M = \struct {A, d}$ be a metric space.

Let $U \subseteq A$.

Then $U$ is an **open set in $M$** if and only if it is a neighborhood of each of its points.

That is:

- $\forall y \in U: \exists \epsilon \in \R_{>0}: \map {B_\epsilon} y \subseteq U$

where $\map {B_\epsilon} y$ is the open $\epsilon$-ball of $y$.

That is, for every point $y$ in $U$, we can find an $\epsilon \in \R_{>0}$, dependent on that $y$, such that the open $\epsilon$-ball of $y$ lies entirely inside $U$.

Another way of saying the same thing is that one can not get out of $U$ by moving an arbitrarily small distance from any point in $U$.

It is important to note that, in general, the values of $\epsilon$ depend on $y$.

That is, it is *not* required that:

- $\exists \epsilon \in \R_{>0}: \forall y \in U: \map {B_\epsilon} y \subseteq U$

### Pseudometric Space

Let $P = \struct {A, d}$ be a pseudometric space.

An **open set** in $P$ is defined in exactly the same way as for a metric space:

$U$ is an **open set in $P$** if and only if:

- $\forall y \in U: \exists \map \epsilon y > 0: \map {B_\epsilon} y \subseteq U$

where $\map {B_\epsilon} y$ is the open $\epsilon$-ball of $y$.

## Also known as

An **open set in $M$** can also be referred to as:

**open in $M$**- a
**$d$-open set** **$d$-open**.

An **open set in $M$** is sometimes seen written as **open subset of $A$**.

## Also see

- Definition:Topology Induced by Metric: the set of all
**open sets**in a given metric space

- Results about
**open sets**can be found**here**.

## Sources

- 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: Compactness - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): $\S 2.6$: Open Sets and Closed Sets: Definition $6.1$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.3$: Open sets in metric spaces: Definition $2.3.8$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**open set**