Is there a name for "almost-subspaces" $U$ that do not satisfy $cu \in U, \forall u \in U$? If so do they have any properties?

Question

I am working with a 'near-subspace' in which the subspace is defined by

$$ U := \{p(x) \in \mathbb{P}^3 \mid p(x)> 0, 0\le x \le 1 \lor p(x) \equiv 0 \} $$

This satisfies that $0\in U$ and it satisfies that $p_1(x) + p_2(x) \in U$, however it does not satisfy the final $cp(x) \in U$ when $c < 0$.

I felt like this was a small thing to miss, and was wondering if there were still any properties that one could gleam from this sort of a subset, such as the existence of a polynomial that is orthogonal to all those in this subset, if there is a basis, or if there was some way to turn this into a vector space