# Definition:Open Set

## Topology

Let $T = \struct {S, \tau}$ be a topological space.

Then the elements of $\tau$ are called the open sets of $T$.

Thus:

$U \in \tau$

and:

$U$ is open in $T$

are equivalent statements.

## Metric Space

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 $y$.

### 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 $y$.

## Normed Vector Space

Let $V = \struct {X, \norm {\,\cdot\,} }$ be a normed vector space.

Let $U \subseteq X$.

Then $U$ is an open set in $V$ if and only if:

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

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

## Complex Analysis

Let $S \subseteq \C$ be a subset of the set of complex numbers.

Let:

$\forall z_0 \in S: \exists \epsilon \in \R_{>0}: N_{\epsilon} \left({z_0}\right) \subseteq S$

where $N_{\epsilon} \left({z_0}\right)$ is the $\epsilon$-neighborhood of $z_0$ for $\epsilon$.

Then $S$ is an open set (of $\C$), or open (in $\C$).

## Real Analysis

Let $I \subseteq \R$ be a subset of the set of real numbers.

Then $I$ is open (in $\R$) if and only if:

$\forall x_0 \in I: \exists \epsilon \in \R_{>0}: \openint {x_0 - \epsilon} {x_0 + \epsilon} \subseteq I$

where $\openint {x_0 - \epsilon} {x_0 + \epsilon}$ is an open interval.

Note that $\epsilon$ may depend on $x_0$.

## Also see

• Results about open sets can be found here.