What is this property called?

Question

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?

Answer 1

Related to absorbing set found in wikipedia

Answer 2

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.

Answer 3

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...