Closure (topology)

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 "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.

DefinitionsEdit

Point of closureEdit

For   a subset of a Euclidean space,   is a point of closure of   if every open ball centered at   contains a point of   (this point may be   itself).

This definition generalizes to any subset   of a metric space   Fully expressed, for   a metric space with metric     is a point of closure of   if for every   there exists some   such that the distance   (again,   is allowed). Another way to express this is to say that   is a point of closure of   if the distance  

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  [1] Note that this definition does not depend upon whether neighbourhoods are required to be open.

Limit pointEdit

The definition of a point of closure 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 limit point, every neighbourhood of the point   in question must contain a point of the set other than   itself. The set of all limit points of a set   is called the derived set of  

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 if there is a neighbourhood of   which contains no other points of   other 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).

Closure of a setEdit

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:

  1.   is the set of all points of closure of  
  2.   is the set   together with all of its limit points.[3]
  3.   is the intersection of all closed sets containing  
  4.   is the smallest closed set containing  
  5.   is the union of   and its boundary  
  6.   is the set of all   for which there exists a net (valued) in   that converges to   in  

The closure of a set has the following properties.[4]

  •   is a closed superset of  
  • The set   is closed if and only if  
  • If   then   is a subset of  
  • 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]

In a first-countable space (such as a metric space),   is the set of all limits of all convergent sequences of points in   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.

ExamplesEdit

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 any space,  
  • In any space    

Giving   and   the standard (metric) topology:

  • If   is the Euclidean space   of real numbers, then  
  • If   is the Euclidean space   then the closure of the set   of rational numbers is the whole space   We say that   is dense in  
  • If   is the complex plane   then  
  • If   is a finite subset of a Euclidean space   then   (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   is endowed with the lower limit topology, then  
  • 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  .

Closure operatorEdit

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:

Theorem[7] (C. Ursescu) — Let   be a sequence of subsets of a complete metric space  

  • If each   is closed in   then
     
  • If each   is open in   then
     

Facts about closuresEdit

A subset   is closed in   if and only if   In particular:

  • The closure of the empty set is the empty set;
  • The closure of   itself is  
  • 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.

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  

Functions and closureEdit

ContinuityEdit

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  

Closed mapsEdit

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 interpretationEdit

One may elegantly define the closure operator in terms of universal arrows, as follows.

The powerset of a set   may be realized as a partial order category   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 alsoEdit

NotesEdit

  1. ^ From   and   it follows that   and   which implies
     

ReferencesEdit

  1. ^ Schubert 1968, p. 20
  2. ^ Kuratowski 1966, p. 75
  3. ^ Hocking & Young 1988, p. 4
  4. ^ Croom 1989, p. 104
  5. ^ Gemignani 1990, p. 55, Pervin 1965, p. 40 and Baker 1991, p. 38 use the second property as the definition.
  6. ^ Pervin 1965, p. 41
  7. ^ Zălinescu 2002, p. 33.

BibliographyEdit

  • Baker, 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 linksEdit