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  2. Empty set - Wikipedia

    en.wikipedia.org/wiki/Empty_set

    In mathematics, the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. [1] Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set , while in other theories, its existence can be deduced.

  3. Lebesgue covering dimension - Wikipedia

    en.wikipedia.org/wiki/Lebesgue_covering_dimension

    The empty set has covering dimension -1: for any open cover of the empty set, each point of the empty set is not contained in any element of the cover, so the order of any open cover is 0. Any given open cover of the unit circle will have a refinement consisting of a collection of open arcs.

  4. Examples of vector spaces - Wikipedia

    en.wikipedia.org/wiki/Examples_of_vector_spaces

    A basis for this vector space is the empty set, so that {0} is the 0-dimensional vector space over F. Every vector space over F contains a subspace isomorphic to this one. The zero vector space is conceptually different from the null space of a linear operator L , which is the kernel of L .

  5. Null set - Wikipedia

    en.wikipedia.org/wiki/Null_set

    The standard construction of the Cantor set is an example of a null uncountable set in ; however other constructions are possible which assign the Cantor set any measure whatsoever. All the subsets of R n {\displaystyle \mathbb {R} ^{n}} whose dimension is smaller than n {\displaystyle n} have null Lebesgue measure in R n . {\displaystyle ...

  6. Lebesgue measure - Wikipedia

    en.wikipedia.org/wiki/Lebesgue_measure

    If a subset of R n has Hausdorff dimension less than n then it is a null set with respect to n-dimensional Lebesgue measure. Here Hausdorff dimension is relative to the Euclidean metric on R n (or any metric Lipschitz equivalent to it). On the other hand, a set may have topological dimension less than n and have positive n-dimensional Lebesgue ...

  7. Axiom of empty set - Wikipedia

    en.wikipedia.org/wiki/Axiom_of_empty_set

    Furthermore, one sometimes considers set theories in which there are no infinite sets, and then the axiom of empty set may still be required. However, any axiom of set theory or logic that implies the existence of any set will imply the existence of the empty set, if one has the axiom schema of separation. This is true, since the empty set is a ...

  8. Nowhere dense set - Wikipedia

    en.wikipedia.org/wiki/Nowhere_dense_set

    The empty set is nowhere dense. In a discrete space, the empty set is the only nowhere dense set. [15] In a T 1 space, any singleton set that is not an isolated point is nowhere dense. A vector subspace of a topological vector space is either dense or nowhere dense. [16]

  9. Image (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Image_(mathematics)

    For the function that maps a Person to their Favorite Food, the image of Gabriela is Apple. The preimage of Apple is the set {Gabriela, Maryam}. The preimage of Fish is the empty set. The image of the subset {Richard, Maryam} is {Rice, Apple}. The preimage of {Rice, Apple} is {Gabriela, Richard, Maryam}.