From planetmath
Let $V$ be a vector space over a field $F$ equipped with a non-discrete valuation $|\cdot|:F\to \mathbb{R}$ . Let $A$ and $B$ be two subsets of $V$. Then $A$ is said to absorb $B$ if there is a non-negative real number $r$ such that, for all $\lambda \in F$ with $|\lambda| \geq r$ , $B \subseteq \lambda A$.
I am thinking about those subsets that can absorb themselves.
- A subset consisting of a single nonzero vector cannot absorb itself, but a subspace can. I was wondering if it is possible to characterize those subsets that can absorb themselves? Must such a subset contain the zero vector? Must it be a subspace?
- Same questions for open subsets in a topological vector space.
Thanks and regards!