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Let $(V,\,+,\,\cdot\,)$ be a vectorspace and $D\subset V$ a set with the following properties

For $\;\lambda D:=\{\lambda d\mid d\in D\}\;\textrm{ and}\;\;\lambda,\, \mu\ge0$: $$0\in D$$ $$\bigcup_{\lambda\in\mathbb{R}}\lambda D=V$$ $$\lambda<\mu\Rightarrow\lambda D\subset\mu D$$ In other words, the set can be bloated up to fill the entire space. It would be practical to give this property a name for the paper I am working on, because it is referenced rather often.

Is there a naming convention? If not, what would you suggest?

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You could describe such a set as $V\backslash \mathbb{R}$, but that is of course a description of the set and not the property... IT also seems to be similar to the concept om hypercyclic operators, but with a set instead of an operator, but I guess that doesn't help either.. :-( – malin Jan 2 '13 at 13:59
@malin indeed it doesnt help ;) Maybe I will just have to invent a name for it, but I want to check if there is a name for it already – CBenni Jan 2 '13 at 14:02
Related to absorbing set as seen here – GEdgar Jan 2 '13 at 14:23
@GEdgar yup, thats what I was looking for. You can post that as an answer ;) – CBenni Jan 2 '13 at 14:25

3 Answers 3

up vote 2 down vote accepted

Related to absorbing set found in wikipedia

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Do you know that $\lambda D\cap\mu D=(0)$ for all $\lambda\neq\mu$? If so, then the name fundamental domain is the most common, since it is almost the fundamental domain of $V$ under the $\mathbb{R}$ action.

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oh, I totally forgot about 1 property >_< sorry. I edited my OP. The sets are subsets of one another – CBenni Jan 2 '13 at 13:54

The trivial subspace $D=\{0\}$ is the name of your set.

For all negative numbers $\lambda$ you have $\lambda D\subseteq 0D=\{0\}$, so $D=\{0\}$

Or did you mean your properties to hold only for non-negative reals?

In case you restrict to non-negative $\lambda$, wouldn't your condition then just mean that $D$ is a neighborhood of the origin? Assuming $V$ is finite dimensional...

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Agreed ... this is not a good quesiton if negative $\lambda$ are allowed. – GEdgar Jan 2 '13 at 14:21
Yes, indeed, I meant only non-negative Integers. – CBenni Jan 2 '13 at 14:24
@ufr, about D being a neighboorhood of the origin, not quite. If D contains a nonempty, open neighborhood of the origin, the properties are fulfilled. GEdgar found that exactly these sets are called "absorbing" – CBenni Jan 2 '13 at 14:49

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