Math for the people, by the people. Encyclopedia .Counterexamples in Topology, Dover, ISBN 048668735X

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    topological space (Definition)

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  • Dense set 1

    Dense setIn topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if any point xin X belongs to A or is a limit point of A.[1] Informally, for every point in X, the point is either in A or arbitrarily"close" to a member of A - for instance, every real number is either a rational number or has one arbitrarily close to it(see Diophantine approximation).Formally, a subset A of a topological space X is dense in X if for any point x in X, any neighborhood of x contains atleast one point from A. Equivalently, A is dense in X if and only if the only closed subset of X containing A is Xitself. This can also be expressed by saying that the closure of A is X, or that the interior of the complement of A isempty.The density of a topological space X is the least cardinality of a dense subset of X.

    Density in metric spacesAn alternative definition of dense set in the case of metric spaces is the following. When the topology of X is givenby a metric, the closure of A in X is the union of A and the set of all limits of sequences of elements in A (its limitpoints),

    Then A is dense in X if

    Note that . If is a sequence of dense open 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.

    ExamplesThe real numbers with the usual topology have the rational numbers as a countable dense subset which shows thatthe cardinality of a dense subset of a topological space may be strictly smaller than the cardinality of the space itself.The irrational numbers are another dense subset which shows that a topological space may have several disjointdense subsets.By the Weierstrass approximation theorem, any given complex-valued continuous function defined on a closedinterval [a,b] can be uniformly approximated as closely as desired by a polynomial function. In other words, thepolynomial functions are dense in the space C[a,b] of continuous complex-valued functions on the interval [a,b],equipped with the supremum norm.Every metric space is dense in its completion.

    PropertiesEvery topological space is dense in itself. For a set X equipped with the discrete topology the whole space is the onlydense set. Every non-empty subset of a set X equipped with the trivial topology is dense, and every topology forwhich every non-empty subset is dense must be trivial.Denseness is transitive: Given three subsets A, B and C of a topological space X with A B C such that A is densein B and B is dense in C (in the respective subspace topology) then A is also dense in C.The image of a dense subset under a surjective continuous function is again dense. The density of a topological spaceis a topological invariant.A topological space with a connected dense subset is necessarily connected itself.

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  • Dense set 2

    Continuous functions into Hausdorff spaces are determined by their values on dense subsets: if two continuousfunctions f, g : X Y into a Hausdorff space Y agree on a dense subset of X then they agree on all of X.

    Related notionsA point x of a subset A of a topological space X is called a limit point of A (in X) if every neighbourhood of x alsocontains a point of A other than x itself, and an isolated point of A otherwise. A subset without isolated points is saidto be dense-in-itself.A subset A of a topological space X is called nowhere dense (in X) if there is no neighborhood in X on which A isdense. Equivalently, a subset of a topological space is nowhere dense if and only if the interior of its closure isempty. The interior of the complement of a nowhere dense set is always dense. The complement of a closed nowheredense set is a dense open set.A topological space with a countable dense subset is called separable. A topological space is a Baire space if andonly if the intersection of countably many dense open sets is always dense. A topological space is called resolvable ifit is the union of two disjoint dense subsets. More generally, a topological space is called -resolvable if it contains pairwise disjoint dense sets.An embedding of a topological space X as a dense subset of a compact space is called a compactification of X.A linear operator between topological vector spaces X and Y is said to be densely defined if its domain is a densesubset of X and if its range is contained within Y. See also continuous linear extension.A topological space X is hyperconnected if and only if every nonempty open set is dense in X. A topological space issubmaximal if and only if every dense subset is open.

    References

    Notes[1] Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, ISBN048668735X

    General references Nicolas Bourbaki (1989) [1971]. General Topology, Chapters 14. Elements of Mathematics. Springer-Verlag.

    ISBN3-540-64241-2. Steen, Lynn Arthur; Seebach, J. Arth