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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.

  1. 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?
  2. Same questions for open subsets in a topological vector space.

Thanks and regards!

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    No. Take the union of a ball with the graph of $\sin{x}$ in $\mathbb{R}^2$. This is my last comment on this thread.2012-01-05

0 Answers 0