Interior Point, Open Set, Closed Set

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Definition (Interior Point, Open Set, Closed Set)

Let $S\subseteq \mathbb{R}$.

• We say that $x$ is an interior point of $S$ if and only if there exists an $\varepsilon>0$ such that $N_\varepsilon(x)\subseteq S$.
• The interior of $S$ is the collection of interior points of $S$.
• We say that $S$ is an open set if and only if for every $x\in S$ there exists an $\varepsilon>0$ such that $N_\varepsilon(x)\subseteq S$ (so every point in $S$ is an interior point, or equivalently $S$ equals the interior of $S$).
• We say that $S$ is a closed set if and only if the complement $\mathbb{R}\setminus S$ is open.