Definition:Open Set
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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 elements.
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.