Definition:Open Set

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Topology

Let $T = \left({S, \tau}\right)$ 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 $M = \left({A, d}\right)$ 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:

$\forall y \in U: \exists \epsilon \in \R_{>0}: B_\epsilon \left({y}\right) \subseteq U$

where $B_\epsilon \left({y}\right)$ is the open $\epsilon$-ball of $y$.


Pseudometric Space

Let $P = \left({A, d}\right)$ 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:

$\forall y \in U: \exists \epsilon \left({y}\right) > 0: B_\epsilon \left({y}\right) \subseteq U$

where $B_\epsilon \left({y}\right)$ is the open $\epsilon$-ball of $y$.


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:

$\forall x_0 \in I: \exists \epsilon \in \R_{>0}: \left({x_0 - \epsilon\,.\,.\,x_0 + \epsilon}\right) \subseteq I$

where $\left({x_0 - \epsilon\,.\,.\,x_0 + \epsilon}\right)$ is an open interval.


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


Also see

  • Results about open sets can be found here.