TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the closure of a subset ''S'' of points in a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
consists of all
point Point or points may refer to: Places * Point, LewisImage:Point Western Isles NASA World Wind.png, Satellite image of Point Point ( gd, An Rubha), also known as the Eye Peninsula, is a peninsula some 11 km long in the Outer Hebrides (or Western I ...

s in ''S'' together with all
limit points In mathematics, a limit point (or cluster point or accumulation point) of a Set (mathematics), set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every Neighbourhood (mathematics), neighbourhood ...
of ''S''. The closure of ''S'' may equivalently be defined as the union of ''S'' and its
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment *Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
, and also as the
intersection The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...
of all
closed set In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
s containing ''S''. Intuitively, the closure can be thought of as all the points that are either in ''S'' or "near" ''S''. A point which is in the closure of ''S'' is a point of closure of ''S''. The notion of closure is in many ways
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality ** . . . see more cases in :Duality theories * Dual ...
to the notion of
interior Interior may refer to: Arts and media * Interior (Degas), ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * Interior (play), ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * The Interior (novel) ...
.

# Definitions

## Point of closure

For $S$ a subset of a
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
, $x$ is a point of closure of $S$ if every
open ball In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
centered at $x$ contains a point of $S$ (this point may be $x$ itself). This definition generalizes to any subset $S$ of a
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
$X.$ Fully expressed, for $X$ a metric space with metric $d,$ $x$ is a point of closure of $S$ if for every $r > 0$ there exists some $s \in S$ such that the distance $d\left(x, s\right) < r$ (again, $x = s$ is allowed). Another way to express this is to say that $x$ is a point of closure of $S$ if the distance $d\left(x, S\right) := \inf_ d\left(x, s\right) = 0.$ This definition generalizes to
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
s by replacing "open ball" or "ball" with "
neighbourhood A neighbourhood (British English British English (BrE) is the standard dialect of the English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval ...

". Let $S$ be a subset of a topological space $X.$ Then $x$ is a or of $S$ if every neighbourhood of $x$ contains a point of $S.$ Note that this definition does not depend upon whether neighbourhoods are required to be open.

## Limit point

The definition of a point of closure is closely related to the definition of a
limit point In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), ''closeness'' is def ...
. The difference between the two definitions is subtle but important – namely, in the definition of limit point, every neighbourhood of the point $x$ in question must contain a point of the set . The set of all limit points of a set $S$ is called the Thus, every limit point is a point of closure, but not every point of closure is a limit point. A point of closure which is not a limit point is an
isolated point 400px, "0" is an isolated point of A = ∪ , 2In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus ...
. In other words, a point $x$ is an isolated point of $S$ if it is an element of $S$ and if there is a neighbourhood of $x$ which contains no other points of $S$ other than $x$ itself. For a given set $S$ and point $x,$ $x$ is a point of closure of $S$ if and only if $x$ is an element of $S$ or $x$ is a limit point of $S$ (or both).

## Closure of a set

The of a subset $S$ of a topological space $\left(X, \tau\right),$ denoted by $\operatorname_ S$ or possibly by $\operatorname_X S$ (if $\tau$ is understood), where if both $X$ and $\tau$ are clear from context then it may also be denoted by $\operatorname S,$ $\overline,$ or $S ^$ (moreover, $\operatorname$ is sometimes capitalized to $\operatorname$) can be defined using any of the following equivalent definitions:
1. $\operatorname S$ is the set of all points of closure of $S.$
2. $\operatorname S$ is the set $S$ together with all of its limit points.
3. $\operatorname S$ is the intersection of all
closed set In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
s containing $S.$
4. $\operatorname S$ is the smallest closed set containing $S.$
5. $\operatorname S$ is the union of $S$ and its
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment *Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
$\partial\left(S\right).$
6. $\operatorname S$ is the set of all $x \in X$ for which there exists a
net Net or net may refer to: Mathematics and physics * Net (mathematics) In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, ...
(valued) in $S$ that converges to $x$ in $\left(X, \tau\right).$
The closure of a set has the following properties. * $\operatorname S$ is a closed superset of $S$ * The set $S$ is closed
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, l ...
$S = \operatorname S$ * If $S \subseteq T$ then $\operatorname S$ is a subset of $\operatorname T.$ * If $A$ is a closed set, then $A$ contains $S$ if and only if $A$ contains $\operatorname S.$ Sometimes the second or third property above is taken as the of the topological closure, which still make sense when applied to other types of closures (see below). In a
first-countable space In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...
(such as a
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
), $\operatorname S$ is the set of all
limits Limit or Limits may refer to: Arts and media * Limit (music) In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre (music), genre of music, or the harmonies that can be made using a particular ...
of all convergent
sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s of points in $S.$ For a general topological space, this statement remains true if one replaces "sequence" by "
net Net or net may refer to: Mathematics and physics * Net (mathematics) In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, ...
" or "
filter Filter, filtering or filters may refer to: Science and technology Device * Filter (chemistry), a device which separates solids from fluids (liquids or gases) by adding a medium through which only the fluid can pass ** Filter (aquarium), critical ...
". Note that these properties are also satisfied if "closure", "superset", "intersection", "contains/containing", "smallest" and "closed" are replaced by "interior", "subset", "union", "contained in", "largest", and "open". For more on this matter, see
closure operatorIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
below.

# Examples

Consider a sphere in 3 dimensions. Implicitly there are two regions of interest created by this sphere; the sphere itself and its interior (which is called an open 3-ball). It is useful to be able to distinguish between the interior of 3-ball and the surface, so we distinguish between the open 3-ball, and the closed 3-ball – the closure of the 3-ball. The closure of the open 3-ball is the open 3-ball plus the surface. In
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
: * In any space, $\varnothing = \operatorname \varnothing.$ * In any space $X,$ $X = \operatorname X.$ Giving $\mathbb$ and $\mathbb$ the standard (metric) topology: * If $X$ is the Euclidean space $\mathbb$ of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, then $\operatorname_X \left(\left(0, 1\right)\right) =$
, 1 The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
* If $X$ is the Euclidean space $\mathbb$ then the closure of the set $\mathbb$ of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s is the whole space $\mathbb.$ We say that $\mathbb$ is
dense The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass Mass is both a property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or ...
in $\mathbb.$ * If $X$ is the
complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

$\mathbb = \mathbb^2,$ then $\operatorname_X \left\left( \ \right\right) = \.$ * If $S$ is a
finite Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
subset of a Euclidean space $X,$ then $\operatorname_X S = S.$ (For a general topological space, this property is equivalent to the T1 axiom.) On the set of real numbers one can put other topologies rather than the standard one. * If $X = \mathbb$ is endowed with the
lower limit topology In mathematics, the lower limit topology or right half-open interval topology is a topological space, topology defined on the set \mathbb of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and h ...
, then * If one considers on $X = \mathbb$ the discrete topology in which every set is closed (open), then $\operatorname_X \left(\left(0, 1\right)\right) = \left(0, 1\right).$ * If one considers on $X = \mathbb$ the trivial topology in which the only closed (open) sets are the empty set and $\mathbb$ itself, then $\operatorname_X \left(\left(0, 1\right)\right) = \mathbb.$ These examples show that the closure of a set depends upon the topology of the underlying space. The last two examples are special cases of the following. * In any
discrete space In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''Isolated point, isolated'' from each other in a certain sense. The discrete topology is t ...
, since every set is closed (and also open), every set is equal to its closure. * In any
indiscrete spaceIn topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, or codiscrete. Intuitively, this has the consequence that a ...
$X,$ since the only closed sets are the empty set and $X$ itself, we have that the closure of the empty set is the empty set, and for every non-empty subset $A$ of $X,$ $\operatorname_X A = X.$ In other words, every non-empty subset of an indiscrete space is
dense The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass Mass is both a property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or ...
. The closure of a set also depends upon in which space we are taking the closure. For example, if $X$ is the set of rational numbers, with the usual
relative topologyIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...

induced by the Euclidean space $\mathbb,$ and if $S = \,$ then $S$ is both closed and open in $\mathbb$ because neither $S$ nor its complement can contain $\sqrt2$, which would be the lower bound of $S$, but cannot be in $S$ because $\sqrt2$ is irrational. So, $S$ has no well defined closure due to boundary elements not being in $\mathbb$. However, if we instead define $X$ to be the set of real numbers and define the interval in the same way then the closure of that interval is well defined and would be the set of all greater than $\sqrt2$.

# Closure operator

A on a set $X$ is a
mapping Mapping may refer to: * Mapping (cartography), the process of making a map * Mapping (mathematics), a synonym for a mathematical function and its generalizations ** Mapping (logic), a synonym for functional predicate Types of mapping * Animated ...
of the
power set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
of $X,$ $\mathcal\left(X\right)$, into itself which satisfies the
Kuratowski closure axioms Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Polish mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...
. Given a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
$\left(X, \tau\right)$, the topological closure induces a function $\operatorname_X : \wp\left(X\right) \to \wp\left(X\right)$ that is defined by sending a subset $S \subseteq X$ to $\operatorname_X S,$ where the notation $\overline$ or $S^$ may be used instead. Conversely, if $\mathbb$ is a closure operator on a set $X,$ then a topological space is obtained by defining the
closed set In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
s as being exactly those subsets $S \subseteq X$ that satisfy $\mathbb\left(S\right) = S$ (so complements in $X$ of these subsets form the
open set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s of the topology). The closure operator $\operatorname_X$ is
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality ** . . . see more cases in :Duality theories * Dual ...
to the
interior Interior may refer to: Arts and media * Interior (Degas), ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * Interior (play), ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * The Interior (novel) ...
operator, which is denoted by $\operatorname_X,$ in the sense that :$\operatorname_X S = X \setminus \operatorname_X \left(X \setminus S\right),$ and also :$\operatorname_X S = X \setminus \operatorname_X \left(X \setminus S\right).$ 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 $X.$ In general, the closure operator does not commute with intersections. However, in a
complete metric space In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathema ...
the following result does hold:

A subset $S$ is closed in $X$ if and only if $\operatorname_X S = S.$ In particular: * The closure of the
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

is the empty set; * The closure of $X$ itself is $X.$ * The closure of an
intersection The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...
of sets is always a
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of (but need not be equal to) the intersection of the closures of the sets. * In a union of
finite Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
ly many sets, the closure of the union and the union of the closures are equal; the union of zero sets is the empty set, and so this statement contains the earlier statement about the closure of the empty set as a special case. * The closure of the union of infinitely many sets need not equal the union of the closures, but it is always a
superset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of the union of the closures. If $S \subseteq T \subseteq X$ and if $T$ is a subspace of $X$ (meaning that $T$ is endowed with the
subspace topologyIn topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a Topological_space#Definitions, topology induced from that of ''X'' called the subspace topology (or the relative ...

that $X$ induces on it), then $\operatorname_T S \subseteq \operatorname_X S$ and the closure of $S$ computed in $T$ is equal to the intersection of $T$ and the closure of $S$ computed in $X$: :$\operatorname_T S ~=~ T \cap \operatorname_X S.$Because $\operatorname_X S$ is a closed subset of $X,$ the intersection $T \cap \operatorname_X S$ is a closed subset of $T$ (by definition of the
subspace topologyIn topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a Topological_space#Definitions, topology induced from that of ''X'' called the subspace topology (or the relative ...

), which implies that $\operatorname_T S \subseteq T \cap \operatorname_X S$ (because $\operatorname_T S$ is the closed subset of $T$ containing $S$). Because $T \cap \operatorname_X S$ is a closed subset of $T,$ from the definition of the subspace topology, there must exist some set $C \subseteq X$ such that $C$ is closed in $X$ and $\operatorname_T S = T \cap C.$ Because $S \subseteq \operatorname_T S \subseteq C$ and $C$ is closed in $X,$ the minimality of $\operatorname_X S$ implies that $\operatorname_X S \subseteq C.$ Intersecting both sides with $T$ shows that $T \cap \operatorname_X S \subseteq T \cap C = \operatorname_T S.$ $\blacksquare$
In particular, $S$ is dense in $T$
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, l ...
$T$ is a subset of $\operatorname_X S.$ If $S, T \subseteq X$ but $S$ is not necessarily a subset of $T$ then only :$\operatorname_T \left(S \cap T\right) ~\subseteq~ T \cap \operatorname_X S$ is guaranteed in general, where this containment could be strict (consider for instance $X = \R$ with the usual topology, $T = \left(-\infty, 0\right],$ and $S = \left(0, \infty\right)$From $T := \left(-\infty, 0\right]$ and $S := \left(0, \infty\right)$ it follows that $S \cap T = \varnothing$ and $\operatorname_X S = \left[0, \infty\right),$ which implies :$\varnothing ~=~ \operatorname_T \left(S \cap T\right) ~\neq~ T \cap \operatorname_X S ~=~ \.$ ) although if $T$ is an open subset of $X$ then the equality $\operatorname_T \left(S \cap T\right) = T \cap \operatorname_X S$ will holdLet $S, T \subseteq X$ and assume that $T$ is open in $X.$ Let $C := \operatorname_T \left(T \cap S\right),$ which is equal to $T \cap \operatorname_X \left(T \cap S\right)$ (because $T \cap S \subseteq T \subseteq X$). The complement $T \setminus C$ is open in $T,$ where $T$ being open in $X$ now implies that $T \setminus C$ is also open in $X.$ Consequently $X \setminus \left(T \setminus C\right) = \left(X \setminus T\right) \cup C$ is a closed subset of $X$ where $\left(X \setminus T\right) \cup C$ contains $S$ as a subset (because if $s \in S$ is in $T$ then $s \in T \cap S \subseteq \operatorname_T \left(T \cap S\right) = C$), which implies that $\operatorname_X S \subseteq \left(X \setminus T\right) \cup C.$ Intersecting both sides with $T$ proves that $T \cap \operatorname_X S \subseteq T \cap C = C.$ The reverse inclusion follows from $C \subseteq \operatorname_X \left(T \cap S\right) \subseteq \operatorname_X S.$ $\blacksquare$ (no matter the relationship between $S$ and $T$). Consequently, if $\mathcal$ is any
open cover In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
of $X$ and if $S \subseteq X$ is any subset then: :$\operatorname_X S = \bigcup_ \operatorname_U \left(U \cap S\right)$ because $\operatorname_U \left(S \cap U\right) = U \cap \operatorname_X S$ for every $U \in \mathcal$ (where every $U \in \mathcal$ is endowed with the
subspace topologyIn topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a Topological_space#Definitions, topology induced from that of ''X'' called the subspace topology (or the relative ...

induced on it by $X$). This equality is particularly useful when $X$ is a
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...
and the sets in the open cover $\mathcal$ are domains of
coordinate chartIn topology, a branch of mathematics, a topological manifold is a topological space which locally resembles real numbers, real ''n''-dimension (mathematics), dimensional Euclidean space. Topological manifolds are an important class of topological spa ...
s. In words, this result shows that the closure in $X$ of any subset $S \subseteq X$ can be computed "locally" in the sets of any open cover of $X$ and then unioned together. In this way, this result can be viewed as the analogue of the well-known fact that a subset $S \subseteq X$ is closed in $X$ if and only if it is "
locally closed This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundame ...
in $X$", meaning that if $\mathcal$ is any
open cover In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
of $X$ then $S$ is closed in $X$ if and only if $S \cap U$ is closed in $U$ for every $U \in \mathcal.$

# Categorical interpretation

One may elegantly define the closure operator in terms of universal arrows, as follows. The
powerset Image:Hasse diagram of powerset of 3.svg, 250px, The elements of the power set of order theory, ordered with respect to Inclusion (set theory), inclusion. In mathematics, the power set (or powerset) of a Set (mathematics), set is the set of al ...

of a set $X$ may be realized as a
partial order upright=1.15, Fig.1 The set of all subsets of a three-element set \, ordered by set inclusion">inclusion Inclusion or Include may refer to: Sociology * Social inclusion, affirmative action to change the circumstances and habits that leads to s ...
category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
$P$ in which the objects are subsets and the morphisms are
inclusion map In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s $A \to B$ whenever $A$ is a subset of $B.$ Furthermore, a topology $T$ on $X$ is a
subcategory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of $P$ with inclusion functor $I : T \to P.$ The set of closed subsets containing a fixed subset $A \subseteq X$ can be identified with the
comma category In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
$\left(A \downarrow I\right).$ This category — also a partial order — then has initial object $\operatorname A.$ Thus there is a universal arrow from $A$ to $I,$ given by the inclusion $A \to \operatorname A.$ Similarly, since every closed set containing $X \setminus A$ corresponds with an open set contained in $A$ we can interpret the category $\left(I \downarrow X \setminus A\right)$ as the set of open subsets contained in $A,$ with
terminal object In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...
$\operatorname\left(A\right),$ the
interior Interior may refer to: Arts and media * Interior (Degas), ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * Interior (play), ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * The Interior (novel) ...
of $A.$ All properties of the closure can be derived from this definition and a few properties of the above categories. Moreover, this definition makes precise the analogy between the topological closure and other types of closures (for example
algebraic closure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
), since all are examples of universal arrows.

* * * * *

* * * * * * * *