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A null set is always considered to be zero dimensional,but a set containing zero vector only is taken to be non-empty set.If something is present there,then why it does'nt make a contribution in part of dimension.

Asaf Karagila
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Kislay Tripathi
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    How is the empty set considerez to be zero-dimensional? It is not a vector space. – Tobias Kildetoft Dec 17 '16 at 13:18
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    The span of the null set is defined to be the zero vector space, which is zero dimensional. The null set is not zero dimensional. – setholopolus Dec 17 '16 at 13:21
  • @TobiasKildetoft why cannot be the empty set be considered a vector space over some arbitrary field? I mean, the axioms that define a vector space, if Im not wrong, holds vacuously. – Masacroso Dec 17 '16 at 13:23
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    They don't. $(\varnothing,+)$ is not an abelian group. – Christoph Dec 17 '16 at 13:24
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    Also note that in the context of affine spaces, $\varnothing$ is an affine space of dimension $-1$, while ${p}$ is an affine space of dimension $0$. – Christoph Dec 17 '16 at 13:25
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    @Masacroso No, they don't hold: the existence of neutral element cannot hold here. – DonAntonio Dec 17 '16 at 13:32
  • If V is a vector space then V itself and 0 vector are always two extreme subspaces of it.Being a subspace means being a vector space itself. – Kislay Tripathi Dec 17 '16 at 14:05

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Note the distinction between saying

  • "The empty set is zero-dimensional" (which is false, as Tobias Kildetoft says, because the empty set does not normally admit the structure of a vector space), and

  • "A zero-dimensional vector space (i.e., a one-element vector space) has the empty set as basis" (which is true, and the reason a one-element vector space is "zero-dimensional").

Andrew D. Hwang
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