Open Set

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In Metric Spaces, a set S is said to be "open" if for every point x in S we can find a small ball centred around x that is entirely inside S.

More formally, $S$ is open if ${\forall}x{\in}S\quad\exists\epsilon{\gt}0\quad:{\quad}d(x,y){\lt}\epsilon\Rightarrow{y}{\in}S.$

The collection of open sets of a metric space, together with the base set, forms a topological space, and the open sets are the topology.