Interior (topology)

See Task:Merge interior page into interior (topology) page - this hasn't been done yet Alec (talk) 19:27, 16 February 2017 (UTC)

Definition

Let $(X,\mathcal{J})$ be a topological space and let $A\in\mathcal{P}(X)$ be an arbitrary subset of $X$, the interior of $A$, with respect to $X$, is denoted and defined as follows:

• $$\text{Int}(A):\eq\bigcup_{U\in\{V\in\mathcal{J}\ \vert\ V\subseteq A\} } U$$ - the interior of $A$ is the union of all open sets contained inside $A$.
  • We use $\text{Int}(A,X)$ to emphasise that we are considering the interior of $A$ with respect to the open sets of $X$.

Equivalent definitions

• $$\text{Int}(A)\eq\bigcup_{x\in\{y\in X\ \vert\ y\text{ is an interior point of }A\} } \{x\}$$
  • See the interior of a set in a topological space is equal to the union of all interior points of that set for proof.

Immediate properties

• $\text{Int}(A)$ is open
  • By definition of $\mathcal{J}$ being a topology it is closed under arbitrary union. The interior is defined to be a union of certain open sets, thus their union is an open set.

See also

• List of topological properties
  • Boundary - denoted $\partial A$
  • Closure - denoted $\overline{A}$

References

Grade: B

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Where did I get the interior point version from? Looking at the interior page (as of now, by ignoring the redirect Alec (talk) 20:10, 16 February 2017 (UTC)) it seems:

• Books:Introduction to Topology - Theodore W. Gamelin & Robert Everist Greene
• Books:Introduction to Topology - Bert Mendelson

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