Definition:Open Set/Metric Space

From ProofWiki
Jump to navigation Jump to search


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

  • Results about open sets can be found here.