Consider a real vector space $V$ with dimension $n$ (say $V=\mathbb{R}^n$). The construction I'm describing is similar to the construction of the projective space. Instead of the space of lines, it is the set of (affine) halfspaces, or equivalently the space of "oriented" (affine) hyperplanes.


Define $A$ the space of non-constant affine functions from $V$ to $\mathbb{R}$. For each $f\in A$ define $S(f)$ as the set of solutions of the equation $f(x)\geq 0$:


$$S(f)=\{x\in V /f(x)\geq 0\}$$


Now give to $S(A)$ the final topology with respect to $S$: open sets of $S(A)$ are those whose preimage by $S$ are open. If you prefer, $S(A)$ is the quotient space of $A$ by the equivalence relation $f\sim g : S(f)=S(g)$.


**Question : describe $S(A)$ topologically (find a "simple" topological space homeomorphic to $S(A)$).**


I'm very interested in special cases esp. $n=2$.



Space of halfspaces
The space is just $\mathbb{R} \times S^{n-1}$: each oriented hyperplane $H$ is identified by the normal unit vector $v$ together with the inner product of $v$ with an arbitrary $w \in H$.


