Closure (topology) - Citizendia

In mathematics, the closure of a set S consists of all points which are intuitively "close to S". Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and This is a glossary of some terms used in the branch of Mathematics known as Topology. 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. In Mathematics, duality has numerous meanings Generally speaking duality is a metamathematical involution. In Mathematics, the interior of a set S consists of all points of S that are intuitively "not on the edge of S "

1 Definitions
2 Examples
3 Closure operator
4 Facts about closures
5 See also

Definitions

Point of closure

For S a subset of an Euclidean space, x is a point of closure of S if every open ball centered at x contains a point of S. In Mathematics, a ball is the inside of a Sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions and for metric (This point may be x itself and x needn't be in S. )

This definition generalises to any subset S of a metric space X. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined Fully expressed, for X a metric space with metric d, x is a point of closure of S if for every r > 0, there is a y in S such that the distance d(x, y) < r. (Again, we may have x = y. ) Another way to express this is to say that x is a point of closure of S if the distance d(x, S) := inf{d(x, s) : s in S} = 0. In Mathematics the infimum of a Subset of some set is the Greatest element, not necessarily in the subset that is less than or equal to all elements of

This definition generalises to topological spaces by replacing "open ball" or "ball" with "neighbourhood". Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. This is a glossary of some terms used in the branch of Mathematics known as Topology. Let S be a subset of a topological space X. Then x is a point of closure 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, informally speaking a limit point of a set S in a Topological space X is a point x in X that can be "approximated" The difference between the two definitions is subtle but important — namely, in the definition of limit point, every neighborhood of the point x in question must contain a point of the set other than x itself.

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 Topology, a branch of Mathematics, a point x of a set S is called an isolated point,if there exists a neighborhood of 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. ↔

Closure of a set

The closure of a set S is the set of all points of closure of S. The closure of S is denoted cl(S), Cl(S), or $\bar{S}$ . The closure of a set has the following properties.

cl(S) is a closed superset of S. In Topology and related branches of Mathematics, a closed set is a set whose complement is open.
cl(S) is the intersection of all closed sets containing S. In Topology and related branches of Mathematics, a closed set is a set whose complement is open.
cl(S) is the smallest closed set containing S.
A set S is closed if and only if S = cl(S). ↔
If S is a subset of T, then cl(S) is a subset of cl(T).
If A is a closed set, then A contains S if and only if A contains cl(S).

Sometimes the second or third property above is taken as the definition of the topological closure.

In a first-countable space (such as a metric space), cl(S) is the set of all limits of all convergent sequences of points in S. In Topology, a branch of Mathematics, a first-countable space is a Topological space satisfying the "first Axiom of countability " In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" In Mathematics, a sequence is an ordered list of objects (or events For a general topological space, this statement remains true if one replaces "sequence" by "net" or "filter". This article is about nets in Topological spaces and not about ε-nets in Metric spaces In Topology and related areas of Mathematics In Mathematics, a filter is a special Subset of a Partially ordered set.

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. In Mathematics, the closure of a set S consists of all points which are intuitively "close to S "

Examples

In any space, the closure of the empty set is the empty set.
In any space X, X = cl(X).
If X is the Euclidean space R of real numbers, then cl((0, 1)) = [0, 1]. In Mathematics, the real numbers may be described informally in several different ways
If X is the Euclidean space R, then the closure of the set Q of rational numbers is the whole space R. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions We say that Q is dense in R. In Topology and related areas of Mathematics, a Subset A of a Topological space X is called dense (in X) if
If X is the complex plane C = R², then cl({z in C : |z| > 1}) = {z in C : |z| ≥ 1}. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted
If S is a finite subset of a Euclidean space, then cl(S) = S. In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. (For a general topological space, this property is equivalent to the T₁ axiom. In Topology and related branches of Mathematics, T1 spaces and R0 spaces are particular kinds of Topological spaces The )

On the set of real numbers one can put other topologies rather than the standard one.

If X = R, where R has the lower limit topology, then cl((0, 1)) = [0, 1). In Mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set R of Real numbers;
If one considers on R the topology in which every set is open (closed), then cl((0, 1)) = (0, 1).
If one considers on R the topology in which the only open (closed) sets are the empty set and R itself, then cl((0, 1)) = 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 open (closed), every set is equal to its closure. In Topology, a discrete space is a particularly simple example of a Topological space or similar structure one in which the points are " isolated "
In any indiscrete space X, since the only open (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, cl(A) = X. In Topology, a Topological space with the trivial topology is one where the only Open sets are the Empty set and the entire space 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 is the set of rational numbers, with the usual subspace topology induced by the Euclidean space R, and if S = {q in Q : q² > 2}, then S is closed in Q, and the closure of S in Q is S; however, the closure of S in the Euclidean space R is the set of all real numbers greater than or equal to $\sqrt2.$

Closure operator

The closure operator ⁻ is dual to the interior operator ^o, in the sense that

S⁻ = X \ (X \ S)^o

and also

S^o = X \ (X \ S)⁻

where X denotes the topological space containing S, and the backslash refers to the set-theoretic difference. In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is In Mathematics, duality has numerous meanings Generally speaking duality is a metamathematical involution. In Mathematics, the interior of a set S consists of all points of S that are intuitively "not on the edge of S " Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Discrete mathematics and predominantly in Set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation

Therefore, the abstract theory of closure operators and the Kuratowski closure axioms can be easily translated into the language of interior operators, by replacing sets with their complements. In Topology and related branches of Mathematics, the Kuratowski closure axioms are a set of Axioms which can be used to define a Topological structure In Discrete mathematics and predominantly in Set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation

Facts about closures

The set $S$ is closed if and only if $C l (S) = S$ . In Topology and related branches of Mathematics, a closed set is a set whose complement is open. In particular, the closure of the empty set is the empty set, and the closure of $X$ itself is $X$ . In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members 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 Mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently 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. In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. 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 $A$ is a subspace of $X$ containing $S$ , then the closure of $S$ computed in $A$ is equal to the intersection of $A$ and the closure of $S$ computed in $X$ : $Cl_A(S) = A\cap Cl_X(S)$ . In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is In particular, $S$ is dense in $A$ iff $A$ is a subset of $C l X (S)$ . ↔

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