# Definition clarification about dimension of a subset

Does it make any sense to talk about the dimension of a subset (not necessarily a subspace) of a vector space at all? What if say you can pick $n$ linearly independent vector from the subset but the set is not closed, then how do we call this number $n$? (Sorry, I know that my question maybe poorly worded, I hope you understand my meaning.) Thanks.

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It sounds like what you're after is the dimension of the span of the subset. Given a vector space $V$ and some subset $S\subseteq V$, not necessarily a subspace of $V$, then $\operatorname{span}(S)$ is the smallest subspace of $V$ that contains $S$. Thus, for example, if one can find $n$ linearly independent vectors in $S$, then we have also found $n$ linearly independent vectors in $\operatorname{span}(S)$ (because $S\subseteq\operatorname{span}(S)$), hence $$\dim(\operatorname{span}(S))\geq n.$$ If $n$ is the largest such number, then in fact we have equality.
For example, the subset $$A = \{(x,y,z)\mid x=0\land y^2+z^2=1\} \subseteq \mathbb R^3$$ spans a plane (of dimension 2), but $A$ is also a manifold of dimension $1$, and if you just speak of the "dimension of $A$", it is likely that people will understand the latter rather than the former.