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Dense set. In topology and related areas of mathematics, a subset A of a 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 are a dense subset of the real numbers because every real number either is a rational number or has a ...
Dense order. In mathematics, a partial order or total order < on a set is said to be dense if, for all and in for which , there is a in such that . That is, for any two elements, one less than the other, there is another element between them. For total orders this can be simplified to "for any two distinct elements, there is another element ...
Dense-in-itself. In general topology, a subset of a topological space is said to be dense-in-itself[1][2] or crowded[3][4] if has no isolated point. Equivalently, is dense-in-itself if every point of is a limit point of . Thus is dense-in-itself if and only if , where is the derived set of . A dense-in-itself closed set is called a perfect set.
Nowhere dense set. In mathematics, a subset of a topological space is called nowhere dense[1][2] or rare[3] if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. For example, the integers are nowhere dense among the reals, whereas the ...
A property is generic in C r if the set holding this property contains a residual subset in the C r topology. Here C r is the function space whose members are continuous functions with r continuous derivatives from a manifold M to a manifold N. The space C r (M, N), of C r mappings between M and N, is a Baire space, hence any residual set is dense.
The definition of a point of closure of a set 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 a limit point of a set , every neighbourhood of must contain a point of other than itself, i.e., each neighbourhood of obviously has but it also must have a point of that is not equal to ...
Dense set A set is dense if it has nonempty intersection with every nonempty open set. Equivalently, a set is dense if its closure is the whole space. Dense-in-itself set A set is dense-in-itself if it has no isolated point. Density the minimal cardinality of a dense subset of a topological space. A set of density ℵ 0 is a separable space ...
Ways of defining sets/Relation to descriptive set theory. Recursive set. Recursively enumerable set. Arithmetical set. Diophantine set. Hyperarithmetical set. Analytical set. Analytic set, Coanalytic set. Suslin set.