{"history":[],"article":{"id":"84fb1863-7336-4833-8b13-9bcfc8c87b13","title":"Dense set","slug":"dense-set","content":":::summary\n:::\n\n# \nIn [topology](https://bliptext.com/articles/topology) and related areas of [mathematics](https://bliptext.com/articles/mathematics), a [subset](https://bliptext.com/articles/subset) A of a [topological space](https://bliptext.com/articles/topological-space) X is said to be **dense** in X if every point of X either belongs to A or else is arbitrarily \"close\" to a member of A — for instance, the [rational numbers](https://bliptext.com/articles/rational-number) are a **dense** [subset](https://bliptext.com/articles/subset) of the [real numbers](https://bliptext.com/articles/real-number) because every real number either is a rational number or has a rational number arbitrarily close to it (see [Diophantine approximation](https://bliptext.com/articles/diophantine-approximation)).\nFormally, A is dense in X if the smallest [closed subset](https://bliptext.com/articles/closed-set) of X containing A is X itself.\n****T****h****e**** **** ****o****f**** ****a**** ****t****o****p****o****l****o****g****i****c****a****l**** ****s****p****a****c****e**** ****X**** ****i****s**** ****t****h****e**** ****l****e****a****s****t**** ****c****a****r****d****i****n****a****l****i****t****y**** ****o****f**** ****a**** ****d****e****n****s****e**** ****s****u****b****s****e****t**** ****o****f**** ****X****.****\n\n# Definition\nA subset A of a [topological space](https://bliptext.com/articles/topological-space) X is said to be a ** of X** if any of the following equivalent conditions are satisfied:\n

The smallest [closed subset](https://bliptext.com/articles/closed-set) of X containing A is X itself.
The [closure](https://bliptext.com/articles/closure-topology) of A in X is equal to X. That is,
The [interior](https://bliptext.com/articles/interior-topology) of the [complement](https://bliptext.com/articles/complement-set-theory) of A is empty.\nThat is,
Every point in X either belongs to A or is a [limit point](https://bliptext.com/articles/limit-point) of A.
For every x \\in X, every [neighborhood](https://bliptext.com/articles/neighbourhood-mathematics) U of x [intersects](https://bliptext.com/articles/intersection-set-theory) A; that is,
A intersects every non-empty open subset of X.

\nand if \\mathcal{B} is a [basis](https://bliptext.com/articles/basis-topology) of open sets for the topology on X then this list can be extended to include:\n

For every x \\in X, every [neighborhood](https://bliptext.com/articles/neighbourhood-mathematics) of x [intersects](https://bliptext.com/articles/intersection-set-theory) A.

A intersects every non-empty

\n\n\n## Density in metric spaces\nAn alternative definition of dense set in the case of [metric spaces](https://bliptext.com/articles/metric-space) is the following.\nWhen the [topology](https://bliptext.com/articles/topology-structure) of X is given by a [metric](https://bliptext.com/articles/metric-mathematics), the [closure](https://bliptext.com/articles/topological-closure) of A in X is the [union](https://bliptext.com/articles/union-set-theory) of A and the set of all [limits of sequences](https://bliptext.com/articles/limit-of-a-sequence) of elements in A (its limit points),\nThen A is dense in X if\nIf is a sequence of dense [open](https://bliptext.com/articles/open-set) 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](https://bliptext.com/articles/baire-category-theorem).\n\n# Examples\nThe [real numbers](https://bliptext.com/articles/real-number) with the usual topology have the [rational numbers](https://bliptext.com/articles/rational-number) as a [countable](https://bliptext.com/articles/countable-set) dense subset which shows that the [cardinality](https://bliptext.com/articles/cardinality) of a dense subset of a topological space may be strictly smaller than the [cardinality](https://bliptext.com/articles/cardinality) of the space itself.\nThe [irrational numbers](https://bliptext.com/articles/irrational-number) are another dense subset which shows that a topological space may have several [disjoint](https://bliptext.com/articles/disjoint-sets) dense subsets (in particular, two dense subsets may be each other's complements), and they need not even be of the same cardinality.\nPerhaps even more surprisingly, both the rationals and the irrationals have empty interiors, showing that dense sets need not contain any non-empty open set.\nThe intersection of two dense open subsets of a topological space is again dense and open.\nThe empty set is a dense subset of itself.\nBut every dense subset of a non-empty space must also be non-empty.\nBy the [Weierstrass approximation theorem](https://bliptext.com/articles/weierstrass-approximation-theorem), any given [complex-valued](https://bliptext.com/articles/complex-number) [continuous function](https://bliptext.com/articles/continuous-function) defined on a [closed interval](https://bliptext.com/articles/closed-interval) [a, b] can be [uniformly approximated](https://bliptext.com/articles/uniform-convergence) as closely as desired by a [polynomial function](https://bliptext.com/articles/polynomial-function).\nIn other words, the polynomial functions are dense in the space C[a, b] of continuous complex-valued functions on the interval [a, b], equipped with the [supremum norm](https://bliptext.com/articles/supremum-norm).\nEvery [metric space](https://bliptext.com/articles/metric-space) is dense in its [completion](https://bliptext.com/articles/completion-metric-space).\n\n# Properties\nEvery [topological space](https://bliptext.com/articles/topological-space) is a dense subset of itself.\nFor a set X equipped with the [discrete topology](https://bliptext.com/articles/discrete-topology), the whole space is the only dense subset.\nEvery non-empty subset of a set X equipped with the [trivial topology](https://bliptext.com/articles/trivial-topology) is dense, and every topology for which every non-empty subset is dense must be trivial.\nDenseness is [transitive](https://bliptext.com/articles/transitive-relation): Given three subsets A, B and C of a topological space X with such that A is dense in B and B is dense in C (in the respective [subspace topology](https://bliptext.com/articles/subspace-topology)) then A is also dense in C.\nThe [image](https://bliptext.com/articles/image-mathematics) of a dense subset under a [surjective](https://bliptext.com/articles/surjective-function) [continuous](https://bliptext.com/articles/continuous-function-topology) function is again dense.\nThe density of a topological space (the least of the [cardinalities](https://bliptext.com/articles/cardinality) of its dense subsets) is a [topological invariant](https://bliptext.com/articles/topological-invariant).\nA topological space with a [connected](https://bliptext.com/articles/connected-space) dense subset is necessarily [connected](https://bliptext.com/articles/connected-space) itself.\nContinuous functions into [Hausdorff spaces](https://bliptext.com/articles/hausdorff-space) are determined by their values on dense subsets: if two continuous functions into a [Hausdorff space](https://bliptext.com/articles/hausdorff-space) Y agree on a dense subset of X then they agree on all of X.\nFor metric spaces there are universal spaces, into which all spaces of given density can be [embedded](https://bliptext.com/articles/embedding): a metric space of density \\alpha is isometric to a subspace of the space of real continuous functions on the [product](https://bliptext.com/articles/cartesian-product) of \\alpha copies of the [unit interval](https://bliptext.com/articles/unit-interval).\n\n# Related notions\nA point x of a subset A of a topological space X is called a [limit point](https://bliptext.com/articles/limit-point) of A (in X) if every neighbourhood of x also contains a point of A other than x itself, and an [isolated point](https://bliptext.com/articles/isolated-point) of A otherwise.\nA subset without isolated points is said to be [dense-in-itself](https://bliptext.com/articles/dense-in-itself).\nA subset A of a topological space X is called [nowhere dense](https://bliptext.com/articles/nowhere-dense-set) (in X) if there is no neighborhood in X on which A is dense.\nEquivalently, a subset of a topological space is nowhere dense if and only if the interior of its closure is empty.\nThe interior of the complement of a nowhere dense set is always dense.\nThe complement of a closed nowhere dense set is a dense open set.\nGiven a topological space X, a subset A of X that can be expressed as the union of countably many nowhere dense subsets of X is called [meagre](https://bliptext.com/articles/meagre-set).\nThe rational numbers, while dense in the real numbers, are meagre as a subset of the reals.\nA topological space with a countable dense subset is called [separable](https://bliptext.com/articles/separable-space).\nA topological space is a [Baire space](https://bliptext.com/articles/baire-space) if and only if the intersection of countably many dense open sets is always dense.\nA topological space is called [resolvable](https://bliptext.com/articles/resolvable-space) if it is the union of two disjoint dense subsets.\nMore generally, a topological space is called κ-resolvable for a [cardinal](https://bliptext.com/articles/cardinal-number) κ if it contains κ pairwise disjoint dense sets.\nAn [embedding](https://bliptext.com/articles/embedding) of a topological space X as a dense subset of a [compact space](https://bliptext.com/articles/compact-space) is called a [compactification](https://bliptext.com/articles/compactification-mathematics) of X.\nA [linear operator](https://bliptext.com/articles/linear-operator) between [topological vector spaces](https://bliptext.com/articles/topological-vector-space) X and Y is said to be [densely defined](https://bliptext.com/articles/densely-defined-operator) if its [domain](https://bliptext.com/articles/domain-of-a-function) is a dense subset of X and if its [range](https://bliptext.com/articles/image-of-a-function) is contained within Y. See also [Continuous linear extension](https://bliptext.com/articles/continuous-linear-extension).\nA topological space X is [hyperconnected](https://bliptext.com/articles/hyperconnected-space) if and only if every nonempty open set is dense in X. A topological space is [submaximal](https://bliptext.com/articles/submaximal-space) if and only if every dense subset is open.\nIf is a metric space, then a non-empty subset Y is said to be \\varepsilon-dense if\nOne can then show that D is dense in if and only if it is ε-dense for every\n\n# General references","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T11:59:33.529Z","updated_at":"2025-01-26T11:59:33.529Z","search_vector":"'dens':1 'set':2","search_content":" \n \n\n# \nIn [topology](https://bliptext.com/articles/topology) and related areas of [mathematics](https://bliptext.com/articles/mathematics), a [subset](https://bliptext.com/articles/subset) A of a [topological space](https://bliptext.com/articles/topological-space) X is said to be **dense** in X if every point of X either belongs to A or else is arbitrarily \"close\" to a member of A — for instance, the [rational numbers](https://bliptext.com/articles/rational-number) are a **dense** [subset](https://bliptext.com/articles/subset) of the [real numbers](https://bliptext.com/articles/real-number) because every real number either is a rational number or has a rational number arbitrarily close to it (see [Diophantine approximation](https://bliptext.com/articles/diophantine-approximation)).\nFormally, A is dense in X if the smallest [closed subset](https://bliptext.com/articles/closed-set) of X containing A is X itself.\n****T****h****e**** **** ****o****f**** ****a**** ****t****o****p****o****l****o****g****i****c****a****l**** ****s****p****a****c****e**** ****X**** ****i****s**** ****t****h****e**** ****l****e****a****s****t**** ****c****a****r****d****i****n****a****l****i****t****y**** ****o****f**** ****a**** ****d****e****n****s****e**** ****s****u****b****s****e****t**** ****o****f**** ****X****.****\n\n# Definition\nA subset A of a [topological space](https://bliptext.com/articles/topological-space) X is said to be a ** of X** if any of the following equivalent conditions are satisfied:\n \n The smallest [closed subset](https://bliptext.com/articles/closed-set) of X containing A is X itself. \n The [closure](https://bliptext.com/articles/closure-topology) of A in X is equal to X. That is, \n The [interior](https://bliptext.com/articles/interior-topology) of the [complement](https://bliptext.com/articles/complement-set-theory) of A is empty.\nThat is, \n Every point in X either belongs to A or is a [limit point](https://bliptext.com/articles/limit-point) of A. \n For every x \\in X, every [neighborhood](https://bliptext.com/articles/neighbourhood-mathematics) U of x [intersects](https://bliptext.com/articles/intersection-set-theory) A; that is, \n A intersects every non-empty open subset of X. \n \n \nand if \\mathcal{B} is a [basis](https://bliptext.com/articles/basis-topology) of open sets for the topology on X then this list can be extended to include:\n For every x \\in X, every [neighborhood](https://bliptext.com/articles/neighbourhood-mathematics) of x [intersects](https://bliptext.com/articles/intersection-set-theory) A. \n A intersects every non-empty \n \n\n## Density in metric spaces\nAn alternative definition of dense set in the case of [metric spaces](https://bliptext.com/articles/metric-space) is the following.\nWhen the [topology](https://bliptext.com/articles/topology-structure) of X is given by a [metric](https://bliptext.com/articles/metric-mathematics), the [closure](https://bliptext.com/articles/topological-closure) of A in X is the [union](https://bliptext.com/articles/union-set-theory) of A and the set of all [limits of sequences](https://bliptext.com/articles/limit-of-a-sequence) of elements in A (its limit points),\nThen A is dense in X if\nIf is a sequence of dense [open](https://bliptext.com/articles/open-set) 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](https://bliptext.com/articles/baire-category-theorem).\n\n# Examples\nThe [real numbers](https://bliptext.com/articles/real-number) with the usual topology have the [rational numbers](https://bliptext.com/articles/rational-number) as a [countable](https://bliptext.com/articles/countable-set) dense subset which shows that the [cardinality](https://bliptext.com/articles/cardinality) of a dense subset of a topological space may be strictly smaller than the [cardinality](https://bliptext.com/articles/cardinality) of the space itself.\nThe [irrational numbers](https://bliptext.com/articles/irrational-number) are another dense subset which shows that a topological space may have several [disjoint](https://bliptext.com/articles/disjoint-sets) dense subsets (in particular, two dense subsets may be each other's complements), and they need not even be of the same cardinality.\nPerhaps even more surprisingly, both the rationals and the irrationals have empty interiors, showing that dense sets need not contain any non-empty open set.\nThe intersection of two dense open subsets of a topological space is again dense and open.\nThe empty set is a dense subset of itself.\nBut every dense subset of a non-empty space must also be non-empty.\nBy the [Weierstrass approximation theorem](https://bliptext.com/articles/weierstrass-approximation-theorem), any given [complex-valued](https://bliptext.com/articles/complex-number) [continuous function](https://bliptext.com/articles/continuous-function) defined on a [closed interval](https://bliptext.com/articles/closed-interval) [a, b] can be [uniformly approximated](https://bliptext.com/articles/uniform-convergence) as closely as desired by a [polynomial function](https://bliptext.com/articles/polynomial-function).\nIn other words, the polynomial functions are dense in the space C[a, b] of continuous complex-valued functions on the interval [a, b], equipped with the [supremum norm](https://bliptext.com/articles/supremum-norm).\nEvery [metric space](https://bliptext.com/articles/metric-space) is dense in its [completion](https://bliptext.com/articles/completion-metric-space).\n\n# Properties\nEvery [topological space](https://bliptext.com/articles/topological-space) is a dense subset of itself.\nFor a set X equipped with the [discrete topology](https://bliptext.com/articles/discrete-topology), the whole space is the only dense subset.\nEvery non-empty subset of a set X equipped with the [trivial topology](https://bliptext.com/articles/trivial-topology) is dense, and every topology for which every non-empty subset is dense must be trivial.\nDenseness is [transitive](https://bliptext.com/articles/transitive-relation): Given three subsets A, B and C of a topological space X with such that A is dense in B and B is dense in C (in the respective [subspace topology](https://bliptext.com/articles/subspace-topology)) then A is also dense in C.\nThe [image](https://bliptext.com/articles/image-mathematics) of a dense subset under a [surjective](https://bliptext.com/articles/surjective-function) [continuous](https://bliptext.com/articles/continuous-function-topology) function is again dense.\nThe density of a topological space (the least of the [cardinalities](https://bliptext.com/articles/cardinality) of its dense subsets) is a [topological invariant](https://bliptext.com/articles/topological-invariant).\nA topological space with a [connected](https://bliptext.com/articles/connected-space) dense subset is necessarily [connected](https://bliptext.com/articles/connected-space) itself.\nContinuous functions into [[Hausdorff space](https://bliptext.com/articles/hausdorff-space)s](https://bliptext.com/articles/hausdorff-space) are determined by their values on dense subsets: if two continuous functions into a [Hausdorff space](https://bliptext.com/articles/hausdorff-space) Y agree on a dense subset of X then they agree on all of X.\nFor metric spaces there are universal spaces, into which all spaces of given density can be [embedded](https://bliptext.com/articles/embedding): a metric space of density \\alpha is isometric to a subspace of the space of real continuous functions on the [product](https://bliptext.com/articles/cartesian-product) of \\alpha copies of the [unit interval](https://bliptext.com/articles/unit-interval).\n\n# Related notions\nA point x of a subset A of a topological space X is called a [limit point](https://bliptext.com/articles/limit-point) of A (in X) if every neighbourhood of x also contains a point of A other than x itself, and an [isolated point](https://bliptext.com/articles/isolated-point) of A otherwise.\nA subset without isolated points is said to be [dense-in-itself](https://bliptext.com/articles/dense-in-itself).\nA subset A of a topological space X is called [nowhere dense](https://bliptext.com/articles/nowhere-dense-set) (in X) if there is no neighborhood in X on which A is dense.\nEquivalently, a subset of a topological space is nowhere dense if and only if the interior of its closure is empty.\nThe interior of the complement of a nowhere dense set is always dense.\nThe complement of a closed nowhere dense set is a dense open set.\nGiven a topological space X, a subset A of X that can be expressed as the union of countably many nowhere dense subsets of X is called [meagre](https://bliptext.com/articles/meagre-set).\nThe rational numbers, while dense in the real numbers, are meagre as a subset of the reals.\nA topological space with a countable dense subset is called [separable](https://bliptext.com/articles/separable-space).\nA topological space is a [Baire space](https://bliptext.com/articles/baire-space) if and only if the intersection of countably many dense open sets is always dense.\nA topological space is called [resolvable](https://bliptext.com/articles/resolvable-space) if it is the union of two disjoint dense subsets.\nMore generally, a topological space is called κ-resolvable for a [cardinal](https://bliptext.com/articles/cardinal-number) κ if it contains κ pairwise disjoint dense sets.\nAn [embedding](https://bliptext.com/articles/embedding) of a topological space X as a dense subset of a [compact space](https://bliptext.com/articles/compact-space) is called a [compactification](https://bliptext.com/articles/compactification-mathematics) of X.\nA [linear operator](https://bliptext.com/articles/linear-operator) between [topological vector spaces](https://bliptext.com/articles/topological-vector-space) X and Y is said to be [densely defined](https://bliptext.com/articles/densely-defined-operator) if its [domain](https://bliptext.com/articles/domain-of-a-function) is a dense subset of X and if its [range](https://bliptext.com/articles/image-of-a-function) is contained within Y. See also [Continuous linear extension](https://bliptext.com/articles/continuous-linear-extension).\nA topological space X is [hyperconnected](https://bliptext.com/articles/hyperconnected-space) if and only if every nonempty open set is dense in X. A topological space is [submaximal](https://bliptext.com/articles/submaximal-space) if and only if every dense subset is open.\nIf is a metric space, then a non-empty subset Y is said to be \\varepsilon-dense if\nOne can then show that D is dense in if and only if it is ε-dense for every\n\n# General references","revisionCount":0},"hasMore":false,"nextCursor":null}

History