In topology, the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S. The closure of S may equivalently be defined as the union of S and its boundary, and also as the intersection of all closed sets containing S. Intuitively, the closure can be thought of as all the points that are either in S or "very 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 to the notion of interior.
For as a subset of a Euclidean space, is a point of closure of if every open ball centered at contains a point of (this point can be itself).
This definition generalizes to any subset of a metric space Fully expressed, for as a metric space with metric is a point of closure of if for every there exists some such that the distance ( is allowed). Another way to express this is to say that is a point of closure of if the distance where is the infimum.
This definition generalizes to topological spaces by replacing "open ball" or "ball" with "neighbourhood". Let be a subset of a topological space Then is a point of closure or adherent point of if every neighbourhood of contains a point of (again, for is allowed).[1] Note that this definition does not depend upon whether neighbourhoods are required to be open.
The definition of a point of closure of a set is closely related to the definition of a limit point of a set. The difference between the two definitions is subtle but important – namely, in the definition of a limit point of a set , every neighbourhood of must contain a point of other than itself, i.e., each neighbourhood of obviously has but it also must have a point of that is not equal to in order for to be a limit point of . A limit point of has more strict condition than a point of closure of in the definitions. The set of all limit points of a set is called the derived set of . A limit point of a set is also called cluster point or accumulation point of the set.
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. In other words, a point is an isolated point of if it is an element of and there is a neighbourhood of which contains no other points of than itself.[2]
For a given set and point is a point of closure of if and only if is an element of or is a limit point of (or both).
The closure of a subset of a topological space denoted by or possibly by (if is understood), where if both and are clear from context then it may also be denoted by or (Moreover, is sometimes capitalized to .) can be defined using any of the following equivalent definitions:
is the set together with all of its limit points. (Each point of is a point of closure of , and each limit point of is also a point of closure of .)[3]
If is a closed set, then contains if and only if contains
Sometimes the second or third property above is taken as the definition of the topological closure, which still make sense when applied to other types of closures (see below).[5]
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 operator below.
Examples
Consider a sphere in a 3 dimensional space. 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 distinguish between the interior and the surface of the sphere, so we distinguish between the open 3-ball (the interior of the sphere), and the closed 3-ball – the closure of the open 3-ball that is the open 3-ball plus the surface (the surface as the sphere itself).
If one considers on the discrete topology in which every set is closed (open), then
If one considers on the trivial topology in which the only closed (open) sets are the empty set and itself, then
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, since every set is closed (and also open), every set is equal to its closure.
In any indiscrete space since the only closed sets are the empty set and itself, we have that the closure of the empty set is the empty set, and for every non-empty subset of In other words, every non-empty subset of an indiscrete space is dense.
The closure of a set also depends upon in which space we are taking the closure. For example, if is the set of rational numbers, with the usual relative topology induced by the Euclidean space and if then is both closed and open in because neither nor its complement can contain , which would be the lower bound of , but cannot be in because is irrational. So, has no well defined closure due to boundary elements not being in . However, if we instead define 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 real numbers greater than or equal to.
A closure operator on a set is a mapping of the power set of , into itself which satisfies the Kuratowski closure axioms. Given a topological space, the topological closure induces a function that is defined by sending a subset to where the notation or may be used instead. Conversely, if is a closure operator on a set then a topological space is obtained by defining the closed sets as being exactly those subsets that satisfy (so complements in of these subsets form the open sets of the topology).[6]
The closure operator is dual to the interior operator, which is denoted by in the sense that
and also
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
In general, the closure operator does not commute with intersections. However, in a complete metric space the following result does hold:
The closure of an intersection of sets is always a subset of (but need not be equal to) the intersection of the closures of the sets.
In a union of finitely 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 of the union of the closures.
Thus, just as the union of two closed sets is closed, so too does closure distribute over binary unions: that is, But just as a union of infinitely many closed sets is not necessarily closed, so too does closure not necessarily distribute over infinite unions: that is, is possible when is infinite.
If and if is a subspace of (meaning that is endowed with the subspace topology that induces on it), then and the closure of computed in is equal to the intersection of and the closure of computed in :
Proof
Because is a closed subset of the intersection is a closed subset of (by definition of the subspace topology), which implies that (because is the smallest closed subset of containing ). Because is a closed subset of from the definition of the subspace topology, there must exist some set such that is closed in and Because and is closed in the minimality of implies that Intersecting both sides with shows that
It follows that is a dense subset of if and only if is a subset of It is possible for to be a proper subset of for example, take and
If but is not necessarily a subset of then only
is always guaranteed, where this containment could be strict (consider for instance with the usual topology, and [proof 1]), although if happens to an open subset of then the equality will hold (no matter the relationship between and ).
Proof
Let and assume that is open in Let which is equal to (because ). The complement is open in where being open in now implies that is also open in Consequently is a closed subset of where contains as a subset (because if is in then ), which implies that Intersecting both sides with proves that The reverse inclusion follows from
Consequently, if is any open cover of and if is any subset then:
because for every (where every is endowed with the subspace topology induced on it by ). This equality is particularly useful when is a manifold and the sets in the open cover are domains of coordinate charts. In words, this result shows that the closure in of any subset can be computed "locally" in the sets of any open cover of and then unioned together. In this way, this result can be viewed as the analogue of the well-known fact that a subset is closed in if and only if it is "locally closed in ", meaning that if is any open cover of then is closed in if and only if is closed in for every
A function between topological spaces is continuous if and only if the preimage of every closed subset of the codomain is closed in the domain; explicitly, this means: is closed in whenever is a closed subset of
In terms of the closure operator, is continuous if and only if for every subset
That is to say, given any element that belongs to the closure of a subset necessarily belongs to the closure of in If we declare that a point is close to a subset if then this terminology allows for a plain English description of continuity: is continuous if and only if for every subset maps points that are close to to points that are close to Thus continuous functions are exactly those functions that preserve (in the forward direction) the "closeness" relationship between points and sets: a function is continuous if and only if whenever a point is close to a set then the image of that point is close to the image of that set. Similarly, is continuous at a fixed given point if and only if whenever is close to a subset then is close to
A function is a (strongly) closed map if and only if whenever is a closed subset of then is a closed subset of In terms of the closure operator, is a (strongly) closed map if and only if for every subset Equivalently, is a (strongly) closed map if and only if for every closed subset
Categorical interpretation
One may define the closure operator in terms of universal arrows, as follows.
The powerset of a set may be realized as a partial ordercategory in which the objects are subsets and the morphisms are inclusion maps whenever is a subset of Furthermore, a topology on is a subcategory of with inclusion functor The set of closed subsets containing a fixed subset can be identified with the comma category This category — also a partial order — then has initial object Thus there is a universal arrow from to given by the inclusion
Similarly, since every closed set containing corresponds with an open set contained in we can interpret the category as the set of open subsets contained in with terminal object the interior of
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), since all are examples of universal arrows.
See also
Adherent point – Point that belongs to the closure of some give subset of a topological space
closure, topology, other, uses, closure, disambiguation, topology, closure, subset, points, topological, space, consists, points, together, with, limit, points, closure, equivalently, defined, union, boundary, also, intersection, closed, sets, containing, intu. For other uses see Closure disambiguation In topology the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S The closure of S may equivalently be defined as the union of S and its boundary and also as the intersection of all closed sets containing S Intuitively the closure can be thought of as all the points that are either in S or very 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 to the notion of interior Contents 1 Definitions 1 1 Point of closure 1 2 Limit point 1 3 Closure of a set 2 Examples 3 Closure operator 4 Facts about closures 5 Functions and closure 5 1 Continuity 5 2 Closed maps 6 Categorical interpretation 7 See also 8 Notes 9 References 10 Bibliography 11 External linksDefinitions EditPoint of closure Edit Main article Adherent point For S displaystyle S as a subset of a Euclidean space x displaystyle x is a point of closure of S displaystyle S if every open ball centered at x displaystyle x contains a point of S displaystyle S this point can be x displaystyle x itself This definition generalizes to any subset S displaystyle S of a metric space X displaystyle X Fully expressed for X displaystyle X as a metric space with metric d displaystyle d x displaystyle x is a point of closure of S displaystyle S if for every r gt 0 displaystyle r gt 0 there exists some s S displaystyle s in S such that the distance d x s lt r displaystyle d x s lt r x s displaystyle x s is allowed Another way to express this is to say that x displaystyle x is a point of closure of S displaystyle S if the distance d x S inf s S d x s 0 displaystyle d x S inf s in S d x s 0 where inf displaystyle inf is the infimum This definition generalizes to topological spaces by replacing open ball or ball with neighbourhood Let S displaystyle S be a subset of a topological space X displaystyle X Then x displaystyle x is a point of closure or adherent point of S displaystyle S if every neighbourhood of x displaystyle x contains a point of S displaystyle S again x s displaystyle x s for s S displaystyle s in S is allowed 1 Note that this definition does not depend upon whether neighbourhoods are required to be open Limit point Edit Main article Limit point of a set The definition of a point of closure of a set is closely related to the definition of a limit point of a set The difference between the two definitions is subtle but important namely in the definition of a limit point x displaystyle x of a set S displaystyle S every neighbourhood of x displaystyle x must contain a point of S displaystyle S other than x displaystyle x itself i e each neighbourhood of x displaystyle x obviously has x displaystyle x but it also must have a point of S displaystyle S that is not equal to x displaystyle x in order for x displaystyle x to be a limit point of S displaystyle S A limit point of S displaystyle S has more strict condition than a point of closure of S displaystyle S in the definitions The set of all limit points of a set S displaystyle S is called the derived set of S displaystyle S A limit point of a set is also called cluster point or accumulation point of the set 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 In other words a point x displaystyle x is an isolated point of S displaystyle S if it is an element of S displaystyle S and there is a neighbourhood of x displaystyle x which contains no other points of S displaystyle S than x displaystyle x itself 2 For a given set S displaystyle S and point x displaystyle x x displaystyle x is a point of closure of S displaystyle S if and only if x displaystyle x is an element of S displaystyle S or x displaystyle x is a limit point of S displaystyle S or both Closure of a set Edit See also Closure mathematics The closure of a subset S displaystyle S of a topological space X t displaystyle X tau denoted by cl X t S displaystyle operatorname cl X tau S or possibly by cl X S displaystyle operatorname cl X S if t displaystyle tau is understood where if both X displaystyle X and t displaystyle tau are clear from context then it may also be denoted by cl S displaystyle operatorname cl S S displaystyle overline S or S displaystyle S Moreover cl displaystyle operatorname cl is sometimes capitalized to Cl displaystyle operatorname Cl can be defined using any of the following equivalent definitions cl S displaystyle operatorname cl S is the set of all points of closure of S displaystyle S cl S displaystyle operatorname cl S is the set S displaystyle S together with all of its limit points Each point of S displaystyle S is a point of closure of S displaystyle S and each limit point of S displaystyle S is also a point of closure of S displaystyle S 3 cl S displaystyle operatorname cl S is the intersection of all closed sets containing S displaystyle S cl S displaystyle operatorname cl S is the smallest closed set containing S displaystyle S cl S displaystyle operatorname cl S is the union of S displaystyle S and its boundary S displaystyle partial S cl S displaystyle operatorname cl S is the set of all x X displaystyle x in X for which there exists a net valued in S displaystyle S that converges to x displaystyle x in X t displaystyle X tau The closure of a set has the following properties 4 cl S displaystyle operatorname cl S is a closed superset of S displaystyle S The set S displaystyle S is closed if and only if S cl S displaystyle S operatorname cl S If S T displaystyle S subseteq T then cl S displaystyle operatorname cl S is a subset of cl T displaystyle operatorname cl T If A displaystyle A is a closed set then A displaystyle A contains S displaystyle S if and only if A displaystyle A contains cl S displaystyle operatorname cl S Sometimes the second or third property above is taken as the definition of the topological closure which still make sense when applied to other types of closures see below 5 In a first countable space such as a metric space cl S displaystyle operatorname cl S is the set of all limits of all convergent sequences of points in S displaystyle S For a general topological space this statement remains true if one replaces sequence by net or filter as described in the article on filters in topology 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 operator below Examples EditConsider a sphere in a 3 dimensional space 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 distinguish between the interior and the surface of the sphere so we distinguish between the open 3 ball the interior of the sphere and the closed 3 ball the closure of the open 3 ball that is the open 3 ball plus the surface the surface as the sphere itself In topological space In any space cl displaystyle varnothing operatorname cl varnothing In other words the closure of the empty set displaystyle varnothing is displaystyle varnothing itself In any space X displaystyle X X cl X displaystyle X operatorname cl X Giving R displaystyle mathbb R and C displaystyle mathbb C the standard metric topology If X displaystyle X is the Euclidean space R displaystyle mathbb R of real numbers then cl X 0 1 0 1 displaystyle operatorname cl X 0 1 0 1 In other words the closure of the set 0 1 displaystyle 0 1 as a subset of X displaystyle X is 0 1 displaystyle 0 1 If X displaystyle X is the Euclidean space R displaystyle mathbb R then the closure of the set Q displaystyle mathbb Q of rational numbers is the whole space R displaystyle mathbb R We say that Q displaystyle mathbb Q is dense in R displaystyle mathbb R If X displaystyle X is the complex plane C R 2 displaystyle mathbb C mathbb R 2 then cl X z C z gt 1 z C z 1 displaystyle operatorname cl X left z in mathbb C z gt 1 right z in mathbb C z geq 1 If S displaystyle S is a finite subset of a Euclidean space X displaystyle X then cl X S S displaystyle operatorname cl 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 R displaystyle X mathbb R is endowed with the lower limit topology then cl X 0 1 0 1 displaystyle operatorname cl X 0 1 0 1 If one considers on X R displaystyle X mathbb R the discrete topology in which every set is closed open then cl X 0 1 0 1 displaystyle operatorname cl X 0 1 0 1 If one considers on X R displaystyle X mathbb R the trivial topology in which the only closed open sets are the empty set and R displaystyle mathbb R itself then cl X 0 1 R displaystyle operatorname cl X 0 1 mathbb R 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 since every set is closed and also open every set is equal to its closure In any indiscrete space X displaystyle X since the only closed sets are the empty set and X displaystyle X itself we have that the closure of the empty set is the empty set and for every non empty subset A displaystyle A of X displaystyle X cl X A X displaystyle operatorname cl X A X In other words every non empty subset of an indiscrete space is dense The closure of a set also depends upon in which space we are taking the closure For example if X displaystyle X is the set of rational numbers with the usual relative topology induced by the Euclidean space R displaystyle mathbb R and if S q Q q 2 gt 2 q gt 0 displaystyle S q in mathbb Q q 2 gt 2 q gt 0 then S displaystyle S is both closed and open in Q displaystyle mathbb Q because neither S displaystyle S nor its complement can contain 2 displaystyle sqrt 2 which would be the lower bound of S displaystyle S but cannot be in S displaystyle S because 2 displaystyle sqrt 2 is irrational So S displaystyle S has no well defined closure due to boundary elements not being in Q displaystyle mathbb Q However if we instead define X displaystyle 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 real numbers greater than or equal to 2 displaystyle sqrt 2 Closure operator EditSee also Closure operator and Kuratowski closure axioms A closure operator on a set X displaystyle X is a mapping of the power set of X displaystyle X P X displaystyle mathcal P X into itself which satisfies the Kuratowski closure axioms Given a topological space X t displaystyle X tau the topological closure induces a function cl X X X displaystyle operatorname cl X wp X to wp X that is defined by sending a subset S X displaystyle S subseteq X to cl X S displaystyle operatorname cl X S where the notation S displaystyle overline S or S displaystyle S may be used instead Conversely if c displaystyle mathbb c is a closure operator on a set X displaystyle X then a topological space is obtained by defining the closed sets as being exactly those subsets S X displaystyle S subseteq X that satisfy c S S displaystyle mathbb c S S so complements in X displaystyle X of these subsets form the open sets of the topology 6 The closure operator cl X displaystyle operatorname cl X is dual to the interior operator which is denoted by int X displaystyle operatorname int X in the sense that cl X S X int X X S displaystyle operatorname cl X S X setminus operatorname int X X setminus S and also int X S X cl X X S displaystyle operatorname int X S X setminus operatorname cl X X setminus S 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 closure operator does not commute with intersections However in a complete metric space the following result does hold Theorem 7 C Ursescu Let S 1 S 2 displaystyle S 1 S 2 ldots be a sequence of subsets of a complete metric space X displaystyle X If each S i displaystyle S i is closed in X displaystyle X then cl X i N int X S i cl X int X i N S i displaystyle operatorname cl X left bigcup i in mathbb N operatorname int X S i right operatorname cl X left operatorname int X left bigcup i in mathbb N S i right right If each S i displaystyle S i is open in X displaystyle X then int X i N cl X S i int X cl X i N S i displaystyle operatorname int X left bigcap i in mathbb N operatorname cl X S i right operatorname int X left operatorname cl X left bigcap i in mathbb N S i right right Facts about closures EditA subset S displaystyle S is closed in X displaystyle X if and only if cl X S S displaystyle operatorname cl X S S In particular The closure of the empty set is the empty set The closure of X displaystyle X itself is X displaystyle X The closure of an intersection of sets is always a subset of but need not be equal to the intersection of the closures of the sets In a union of finitely 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 of the union of the closures Thus just as the union of two closed sets is closed so too does closure distribute over binary unions that is cl X S T cl X S cl X T displaystyle operatorname cl X S cup T operatorname cl X S cup operatorname cl X T But just as a union of infinitely many closed sets is not necessarily closed so too does closure not necessarily distribute over infinite unions that is cl X i I S i i I cl X S i displaystyle operatorname cl X left bigcup i in I S i right neq bigcup i in I operatorname cl X S i is possible when I displaystyle I is infinite If S T X displaystyle S subseteq T subseteq X and if T displaystyle T is a subspace of X displaystyle X meaning that T displaystyle T is endowed with the subspace topology that X displaystyle X induces on it then cl T S cl X S displaystyle operatorname cl T S subseteq operatorname cl X S and the closure of S displaystyle S computed in T displaystyle T is equal to the intersection of T displaystyle T and the closure of S displaystyle S computed in X displaystyle X cl T S T cl X S displaystyle operatorname cl T S T cap operatorname cl X S ProofBecause cl X S displaystyle operatorname cl X S is a closed subset of X displaystyle X the intersection T cl X S displaystyle T cap operatorname cl X S is a closed subset of T displaystyle T by definition of the subspace topology which implies that cl T S T cl X S displaystyle operatorname cl T S subseteq T cap operatorname cl X S because cl T S displaystyle operatorname cl T S is the smallest closed subset of T displaystyle T containing S displaystyle S Because cl T S displaystyle operatorname cl T S is a closed subset of T displaystyle T from the definition of the subspace topology there must exist some set C X displaystyle C subseteq X such that C displaystyle C is closed in X displaystyle X and cl T S T C displaystyle operatorname cl T S T cap C Because S cl T S C displaystyle S subseteq operatorname cl T S subseteq C and C displaystyle C is closed in X displaystyle X the minimality of cl X S displaystyle operatorname cl X S implies that cl X S C displaystyle operatorname cl X S subseteq C Intersecting both sides with T displaystyle T shows that T cl X S T C cl T S displaystyle T cap operatorname cl X S subseteq T cap C operatorname cl T S displaystyle blacksquare It follows that S T displaystyle S subseteq T is a dense subset of T displaystyle T if and only if T displaystyle T is a subset of cl X S displaystyle operatorname cl X S It is possible for cl T S T cl X S displaystyle operatorname cl T S T cap operatorname cl X S to be a proper subset of cl X S displaystyle operatorname cl X S for example take X R displaystyle X mathbb R S 0 1 displaystyle S 0 1 and T 0 displaystyle T 0 infty If S T X displaystyle S T subseteq X but S displaystyle S is not necessarily a subset of T displaystyle T then onlycl T S T T cl X S displaystyle operatorname cl T S cap T subseteq T cap operatorname cl X S is always guaranteed where this containment could be strict consider for instance X R displaystyle X mathbb R with the usual topology T 0 displaystyle T infty 0 and S 0 displaystyle S 0 infty proof 1 although if T displaystyle T happens to an open subset of X displaystyle X then the equality cl T S T T cl X S displaystyle operatorname cl T S cap T T cap operatorname cl X S will hold no matter the relationship between S displaystyle S and T displaystyle T ProofLet S T X displaystyle S T subseteq X and assume that T displaystyle T is open in X displaystyle X Let C cl T T S displaystyle C operatorname cl T T cap S which is equal to T cl X T S displaystyle T cap operatorname cl X T cap S because T S T X displaystyle T cap S subseteq T subseteq X The complement T C displaystyle T setminus C is open in T displaystyle T where T displaystyle T being open in X displaystyle X now implies that T C displaystyle T setminus C is also open in X displaystyle X Consequently X T C X T C displaystyle X setminus T setminus C X setminus T cup C is a closed subset of X displaystyle X where X T C displaystyle X setminus T cup C contains S displaystyle S as a subset because if s S displaystyle s in S is in T displaystyle T then s T S cl T T S C displaystyle s in T cap S subseteq operatorname cl T T cap S C which implies that cl X S X T C displaystyle operatorname cl X S subseteq X setminus T cup C Intersecting both sides with T displaystyle T proves that T cl X S T C C displaystyle T cap operatorname cl X S subseteq T cap C C The reverse inclusion follows from C cl X T S cl X S displaystyle C subseteq operatorname cl X T cap S subseteq operatorname cl X S displaystyle blacksquare Consequently if U displaystyle mathcal U is any open cover of X displaystyle X and if S X displaystyle S subseteq X is any subset then cl X S U U cl U U S displaystyle operatorname cl X S bigcup U in mathcal U operatorname cl U U cap S because cl U S U U cl X S displaystyle operatorname cl U S cap U U cap operatorname cl X S for every U U displaystyle U in mathcal U where every U U displaystyle U in mathcal U is endowed with the subspace topology induced on it by X displaystyle X This equality is particularly useful when X displaystyle X is a manifold and the sets in the open cover U displaystyle mathcal U are domains of coordinate charts In words this result shows that the closure in X displaystyle X of any subset S X displaystyle S subseteq X can be computed locally in the sets of any open cover of X displaystyle 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 X displaystyle S subseteq X is closed in X displaystyle X if and only if it is locally closed in X displaystyle X meaning that if U displaystyle mathcal U is any open cover of X displaystyle X then S displaystyle S is closed in X displaystyle X if and only if S U displaystyle S cap U is closed in U displaystyle U for every U U displaystyle U in mathcal U Functions and closure EditContinuity Edit Main article Continuous function A function f X Y displaystyle f X to Y between topological spaces is continuous if and only if the preimage of every closed subset of the codomain is closed in the domain explicitly this means f 1 C displaystyle f 1 C is closed in X displaystyle X whenever C displaystyle C is a closed subset of Y displaystyle Y In terms of the closure operator f X Y displaystyle f X to Y is continuous if and only if for every subset A X displaystyle A subseteq X f cl X A cl Y f A displaystyle f left operatorname cl X A right subseteq operatorname cl Y f A That is to say given any element x X displaystyle x in X that belongs to the closure of a subset A X displaystyle A subseteq X f x displaystyle f x necessarily belongs to the closure of f A displaystyle f A in Y displaystyle Y If we declare that a point x displaystyle x is close to a subset A X displaystyle A subseteq X if x cl X A displaystyle x in operatorname cl X A then this terminology allows for a plain English description of continuity f displaystyle f is continuous if and only if for every subset A X displaystyle A subseteq X f displaystyle f maps points that are close to A displaystyle A to points that are close to f A displaystyle f A Thus continuous functions are exactly those functions that preserve in the forward direction the closeness relationship between points and sets a function is continuous if and only if whenever a point is close to a set then the image of that point is close to the image of that set Similarly f displaystyle f is continuous at a fixed given point x X displaystyle x in X if and only if whenever x displaystyle x is close to a subset A X displaystyle A subseteq X then f x displaystyle f x is close to f A displaystyle f A Closed maps Edit Main article Open and closed maps A function f X Y displaystyle f X to Y is a strongly closed map if and only if whenever C displaystyle C is a closed subset of X displaystyle X then f C displaystyle f C is a closed subset of Y displaystyle Y In terms of the closure operator f X Y displaystyle f X to Y is a strongly closed map if and only if cl Y f A f cl X A displaystyle operatorname cl Y f A subseteq f left operatorname cl X A right for every subset A X displaystyle A subseteq X Equivalently f X Y displaystyle f X to Y is a strongly closed map if and only if cl Y f C f C displaystyle operatorname cl Y f C subseteq f C for every closed subset C X displaystyle C subseteq X Categorical interpretation EditOne may define the closure operator in terms of universal arrows as follows The powerset of a set X displaystyle X may be realized as a partial order category P displaystyle P in which the objects are subsets and the morphisms are inclusion maps A B displaystyle A to B whenever A displaystyle A is a subset of B displaystyle B Furthermore a topology T displaystyle T on X displaystyle X is a subcategory of P displaystyle P with inclusion functor I T P displaystyle I T to P The set of closed subsets containing a fixed subset A X displaystyle A subseteq X can be identified with the comma category A I displaystyle A downarrow I This category also a partial order then has initial object cl A displaystyle operatorname cl A Thus there is a universal arrow from A displaystyle A to I displaystyle I given by the inclusion A cl A displaystyle A to operatorname cl A Similarly since every closed set containing X A displaystyle X setminus A corresponds with an open set contained in A displaystyle A we can interpret the category I X A displaystyle I downarrow X setminus A as the set of open subsets contained in A displaystyle A with terminal object int A displaystyle operatorname int A the interior of A displaystyle 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 since all are examples of universal arrows See also EditAdherent point Point that belongs to the closure of some give subset of a topological space Closure algebra Closed regular set a set equal to the closure of their interior Derived set mathematics set of all limit points of a set S is called the derived set of SPages displaying wikidata descriptions as a fallback Interior topology Largest open subset of some given set Limit point of a set Cluster point in a topological spacePages displaying short descriptions of redirect targetsNotes Edit From T 0 displaystyle T infty 0 and S 0 displaystyle S 0 infty it follows that S T displaystyle S cap T varnothing and cl X S 0 displaystyle operatorname cl X S 0 infty which implies cl T S T T cl X S 0 displaystyle varnothing operatorname cl T S cap T neq T cap operatorname cl X S 0 References Edit Schubert 1968 p 20 Kuratowski 1966 p 75 Hocking amp Young 1988 p 4 Croom 1989 p 104 Gemignani 1990 p 55 Pervin 1965 p 40 and Baker 1991 p 38 use the second property as the definition Pervin 1965 p 41 Zălinescu 2002 p 33 Bibliography EditBaker Crump W 1991 Introduction to Topology Wm C Brown Publisher ISBN 0 697 05972 3 Croom Fred H 1989 Principles of Topology Saunders College Publishing ISBN 0 03 012813 7 Gemignani Michael C 1990 1967 Elementary Topology 2nd ed Dover ISBN 0 486 66522 4 Hocking John G Young Gail S 1988 1961 Topology Dover ISBN 0 486 65676 4 Kuratowski K 1966 Topology vol I Academic Press Pervin William J 1965 Foundations of General Topology Academic Press Schubert Horst 1968 Topology Allyn and Bacon Zălinescu Constantin 30 July 2002 Convex Analysis in General Vector Spaces River Edge N J London World Scientific Publishing ISBN 978 981 4488 15 0 MR 1921556 OCLC 285163112 via Internet Archive External links Edit Closure of a set Encyclopedia of Mathematics EMS Press 2001 1994 Retrieved from https en wikipedia org w index php title Closure topology amp oldid 1140612720, wikipedia, wiki, book, books, library,