Open Set (Topology)

In topology

In a topological space $X$, open sets are basic concepts defined axiomatically as subsets of the space. The set of open sets $T$ is closed on:

• $\varnothing \in T$ and $X\in T$
• $\bigcap S\in T$ for $S\subseteq T$, $|S|$ countable.
• $\bigcup S\in T$ for $S\subseteq T$ ($|S|$ arbitrary).

In Metric Space

In a metric space, a set $X\subseteq Y$ is open iff For every point $x\in X$, there is some radius $ϵ\in \mathbb{R}$ such that the open ball of radius $ϵ$ at $x$, $B_{ϵ}(x)\subset X$.