Category:Open Sets (Metric Spaces)

This category contains results about Open Sets in the context of Metric Spaces.

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 $y$.