# Interior (topology)

(Redirected from Interior)
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 [ilmath](X,\mathcal{J})[/ilmath] be a topological space and let [ilmath]A\in\mathcal{P}(X)[/ilmath] be an arbitrary subset of [ilmath]X[/ilmath], the interior of [ilmath]A[/ilmath], with respect to [ilmath]X[/ilmath], is denoted and defined as follows[1]:

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

## Immediate properties

• [ilmath]\text{Int}(A)[/ilmath] is open
• By definition of [ilmath]\mathcal{J} [/ilmath] 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.