Interior Point

Topics: Calculus - Topology

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(definition)

Let $x\in A\subseteq \mathbb{R}$. We say that $x$ is an interior point of $A$ if $\exists \delta \in {\mathbb{R}}^{+}$ such that $B_{\delta }(x)\subseteq A$ (the open ball).

In other words, $x$ is an interior point if we can form an open ball such that all of the ball is contained within $A$.

The set of interior points of $A$ is called the interior of $A$, written $\text{int}(A)$.

Example

Let $A=[-1,1]$.

• $x=0$ is an interior point, since $\delta \in [0,1]$ yields a ball that’s contained in $A$.
• $x=-1$ is not an interior point, since there’s no $\delta$ such that the resulting ball is contained in $A$.

If $A=\text{int}(A)$, then $A$ is an open set.