“But it is not only of space in the Church which we ought to be jealous, but also of the interior of the house of God in us, so that it might not become a house of merchandise, or a den of robbers.”
— Ambrose, Commentary on John 2:16, Exposition of the Psalms of David 118 (PL 15 1457B)
“The trick is to maintain a kind of naïve amazement at each instant of experience—but, as Montaigne learned, one of the best techniques for doing this is to write about everything. Simply describing an object on your table, or the view from your window, opens your eyes to how marvelous such ordinary things are. To look inside yourself is to open up an even more fantastical realm.”
— Sarah Bakewell, How to Live (2010), p. 37
“Where there have been powerful governments, societies, religions, public opinions, in short wherever there has been tyranny, there the solitary philosopher has been hated; for philosophy offers an asylum to a man into which no tyranny can force it way, the inward cave, the labyrinth of the heart.”
— Friedrich Nietzsche, Untimely Meditations, trans. Hollingdale, “Schopenhauer as educator,” § 3.3, p. 139
“Everything will be dead inside you. Never again will you be capable of love, or friendship, or joy of living, or laughter, or curiosity, or courage, or integrity. You will be hollow. We shall squeeze you empty and then we shall fill you with ourselves.”
— George Orwell, Winston Smith's torturer, 1984
“What is necessary, after all, is only this: solitude, vast inner solitude. To walk inside yourself and meet no one for hours—that is what you must be able to attain.”
— Rainer Maria Rilke, Letters to a Young Poet, 6
“Be at least as interested in what goes on inside you as what happens outside.”
— Eckhart Tolle, The Power of Now
Quotes via Wikiquote (CC BY-SA), each with its original source.
Article
In mathematics, specifically in topology,
the interior of a subset S of a topological space X is the union of all subsets of S that are open in X.
A point that is in the interior of S is an interior point of S.
The interior of S is the complement of the closure of the complement of S.
In this sense interior and closure are dual notions.
The exterior of a set S is the complement of the closure of S; it consists of the points that are in neither the set nor its boundary.
The interior, boundary, and exterior of a subset together partition the whole space into three blocks (or fewer when one or more of these is empty).
Contents
Definitions
Interior point
If
S
{\displaystyle S}
is a subset of a Euclidean space, then
x
{\displaystyle x}
is an interior point of
S
{\displaystyle S}
if there exists an open ball centered at
x
{\displaystyle x}
which is completely contained in
S
.
{\displaystyle S.}
(This is illustrated in the introductory section to this article.)
This definition generalizes to any subset
S
{\displaystyle S}
of a metric space
X
{\displaystyle X}
with metric
d
{\displaystyle d}
:
x
{\displaystyle x}
Interior of a set
The interior of a subset
S
{\displaystyle S}
of a topological space
X
,
{\displaystyle X,}
denoted by
int
X
S
{\displaystyle \operatorname {int} _{X}S}
or
int
S
{\displaystyle \operatorname {int} S}
or
S
∘
,
{\displaystyle S^{\circ },}
can be defined in any of the following equivalent ways:
int
S
{\displaystyle \operatorname {int} S}
is the largest open subset of
X
{\displaystyle X}
Examples
In any space, the interior of the empty set is the empty set.
In any space
X
,
{\displaystyle X,}
if
S
⊆
X
,
{\displaystyle S\subseteq X,}
then
int
S
⊆
S
.
{\displaystyle \operatorname {int} S\subseteq S.}
If
X
{\displaystyle X}
is the real line
R
{\displaystyle \mathbb {R} }
(with the standard topology), then
int
(
[
0
,
Properties
Let
X
{\displaystyle X}
be a topological space and let
S
{\displaystyle S}
and
T
{\displaystyle T}
be subsets of
X
.
{\displaystyle X.}
int
S
{\displaystyle \operatorname {int} S}
is open in
X
.
{\displaystyle X.}
If
T
{\displaystyle T}
is open in
X
{\displaystyle X}
then
T
⊆
S
{\displaystyle T\subseteq S}
if and only if
Interior operator
The interior operator
int
X
{\displaystyle \operatorname {int} _{X}}
is dual to the closure operator, which is denoted by
Therefore, the abstract theory of closure operators and the Kuratowski closure axioms can be readily translated into the language of interior operators, by replacing sets with their complements in
X
.
{\displaystyle X.}
In general, the interior operator does not commute with unions. However, in a complete metric space the following result does hold:
The result above implies that every complete metric space is a Baire space.
The exterior is the interior of the complement, which is the same as the complement of the closure; in formulas,
