Interior

Definition

Given a set $U\subseteq X$ and an arbitrary metric space, $(X,d)$ or topological space, $(X,\mathcal{J})$ the interior of $U$, denoted $\text{Int}(U)$ is defined as:

• $\text{Int}(U):=\{x\in X\vert\ x\text{ is interior to }U\}$ - (see interior point for the definition of what it means to be interior to)

Note that, unlike interior point which is basically a synonym for neighbourhood (taking the definition of neighbourhood as discussed on its page) the interior is a meaningful and distinct definition. In accordance with the topological definition of interior point (requiring that $U$ be a neighbourhood to some $x\in X$) we see that:

• $\text{Int}(U)$ is the set of all points $U$ is a neighbourhood to.

Immediate properties

Let $U\subseteq X$ be an arbitrary subset of a topological space $(X,\mathcal{J})$ (as all metric spaces are topological, they are included), then:

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Claim 1: $\text{Int}(U)$ is open

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Recall that for $U$ to be open we require that $\text{Int}(U)=U$, so for $\text{Int}(U)$ to be open we require that:

• $\text{Int}(\text{Int}(U))=\text{Int}(U)$

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TODO: Be bothered to fill in this proof

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See also

• Open set
• Interior point
• Topological space