Definition:Open Set/Metric Space

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Definition

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 $y$.


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:

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


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 $y$.


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

  • Results about open sets can be found here.


Sources