Let be the set of non-zero polynomials $P(x)$ of degree at most two with constant coefficients in $\mathbb{R}$ and a single variable $x$ and possessing real zeros. Given a polynomial $P(x)\equiv{}a\,x^2+b\,x+c$, then the real zeros of this polynomial, when they exist, are the same as those of polynomials $Q(x)=x^2+u\,x+v$ where $u=b/a$ and $v=c/a$ if $a\neq0$, or of polynomials $L(x)=x+w$ if $a=0$. In the latter case, these polynomials $L(x)$ can be considered the limit polynomials of polynomials $Q(x)$ when $(u,v)\to(\infty,\infty)$ and $v/u\to{w}$. Also, among these polynomials $L(x)$, we have those of the factorizable polynomials $Q(x)$ possessing a single zero and therefore such that $u^2=4\,v$. Finally, the set $\mathcal{K}$ of constant polynomials $K(x)=c\neq0$ can be considered as the limit set of the set of polynomials $Q(x)$ when $(u,v)\to(\infty,\infty)$ and $u/v\to0$, i.e., also when $w\to\infty$. Thus, when $(u,v)\to(\infty,\infty)$ and $|w|<\infty$, we get polyomials $L(x)$ as limits of polynomials $Q(x)$ whereas if $(u,v)\to(\infty,\infty)$ and $w\equiv\infty$, we get a limit set $\mathcal{K}$ as limit of a set.
Then the question is: can we associate the limit set $\mathcal{K}$ of non-zero constant polynomials with a point at infinity $\omega$ and then form the Alexandroff extension $\mathcal{A}\equiv\mathbb{R}^2\cup\{\omega\}\equiv{P}\mathbb{R}^2\equiv\mathbb{R}^2\cup{P}\mathbb{R}^1$ of $\mathbb{R}^2$ constituting the set of polynomials of degree two with real coefficients? In principle, yes, since all we need is a locally compact space, but... is that enough in this case and do we really get ${P}\mathbb{R}^2$?
