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Dense set
 
 
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In topology
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
 and related areas of mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a subset
Subset

In mathematics, especially in set theory, a Set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide....
 A of a topological space
Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
 X is called dense (in X) if, intuitively, any point in X can be "well-approximated" by points in A. Formally, A is dense in X if for any point x in X, any neighborhood of x contains at least one point from A.

Equivalently, A is dense in X if the only closed subset
Closed set

In topology and related branches of mathematics, a closed set is a Set whose complement is open set....
 of X containing A is X itself.






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Encyclopedia


In topology
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
 and related areas of mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a subset
Subset

In mathematics, especially in set theory, a Set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide....
 A of a topological space
Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
 X is called dense (in X) if, intuitively, any point in X can be "well-approximated" by points in A. Formally, A is dense in X if for any point x in X, any neighborhood of x contains at least one point from A.

Equivalently, A is dense in X if the only closed subset
Closed set

In topology and related branches of mathematics, a closed set is a Set whose complement is open set....
 of X containing A is X itself. This can also be expressed by saying that the closure
Closure (topology)

In mathematics, the closure of a set S consists of all Topology glossary#Ps which are intuitively "close to S". A point which is in the closure of S is a adherent point of S....
 of A is X, or that the interior
Interior (topology)

In mathematics, the interior of a set S consists of all Topology glossary#Ps of S that are intuitively "not on the edge of S". A point that is in the interior of S is an interior point of S....
 of the complement of A is empty.

Density in metric spaces


An alternative definition of dense set in the case of metric space
Metric space

In mathematics, a metric space is a Set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space....
s is the following: The set A in a metric space X is dense if every x in X is a limit of a sequence
Limit of a sequence

The limit of a sequence is one of the oldest concepts in mathematical analysis. It provides a rigorous definition of the idea of a sequence converging towards a point called the limit....
 of elements in A. Indeed, when the topology of X is given by a metric, the closure
Closure (mathematics)

In mathematics, a Set is said to be closed under some operation if the Operation on members of the set produces a member of the set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 7 are both natural numbers, but the result of 3 − 7 is not....
  of A in X is the set of all limits of sequences of elements in A,


If is a sequence of dense open
Open set

In metric topology and related fields of mathematics, a Set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U....
 sets in a complete metric space, X, then is also dense in X. This fact is one of the equivalent forms of the Baire category theorem
Baire category theorem

The Baire category theorem is an important tool in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space....
.

Examples


  • Every topological space
    Topological space

    Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
     is dense in itself.
  • The real number
    Real number

    In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
    s with the usual topology have the rational number
    Rational number

    In mathematics, a rational number is a number which can be expressed as a quotient of two integers. Non-integer rational numbers are usually written as the vulgar fraction , where b is not 0 ....
    s and the irrational number
    Irrational number

    In mathematics, an irrational number is any real number that is not a rational number ? that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero....
    s as dense subsets.
  • A metric space
    Metric space

    In mathematics, a metric space is a Set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space....
      is dense in its completion .


See also


  • Dense order
    Dense order

    In mathematics, a partial order ≤ on a set X is said to be dense if, for all x and y in X for which x < y, there is a z in X such that x < z < y....
  • Dense-in-itself
    Dense-in-itself

    In mathematics, a subset of a topological space is said to be dense-in-itself if contains no isolated points.Note that if the subset is also a closed set, then will be a perfect set....
  • Separable space
    Separable space

    In mathematics a topological space is called separable if it contains a countable set dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence....
    , a space with a countable dense subset
  • Nowhere dense set
    Nowhere dense set

    In topology, a subset A of a topological space X is called nowhere dense if the interior of the closure of A is empty set. For example, the integers form a nowhere dense subset of the real line R....
    , the opposite notion