Interior (topology)

From Maths
(Redirected from Interior)
Jump to: navigation, search
See Task:Merge interior page into interior (topology) page - this hasn't been done yet Alec (talk) 19:27, 16 February 2017 (UTC)


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]:

  • [math]\text{Int}(A):\eq\bigcup_{U\in\{V\in\mathcal{J}\ \vert\ V\subseteq A\} } U[/math] - 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].

Equivalent definitions

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.

See also


  1. see interior point (topology) as needed for definition


  1. Introduction to Topological Manifolds - John M. Lee
Grade: B
This page requires references, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.
The message provided is:
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: have something to say