It is nice that you asked a question about the space $\mathbb Q P^\infty$. I have thought about this space for a long time and came to the conclusion that $\mathbb Q P^\infty$ is the most "regular" space among countable connected Hausdorff spaces. It seems that $\mathbb Q P^\infty$ is a unique space among countable connected Hausdorff spaces that admits a simle topological characterization:

Theorem. A topological space $X$ is homeomorphic to $\mathbb QP^\infty$ if and only if $X$ is countable, Hausdorff, and has a countable base $\mathcal B$ of the topology such that for any $n\ge 2$ and basic open sets $U_1,\dots,U_n\in\mathcal B$ the intersection $\bar U_1\cap\dots\cap \bar U_n$ is connected, non-empty, and has zero-dimensional complement $X\setminus (\bar U_1\cap\dots\cap \bar U_n)$.

It is nice that you asked a question about the space $\mathbb Q P^\infty$. I have thought about this space for a long time and came to the conclusion that $\mathbb Q P^\infty$ is the most "regular" space among countable connected Hausdorff spaces. It seems that $\mathbb Q P^\infty$ is a unique space among countable connected Hausdorff spaces that admits a simle topological characterization:

Theorem. A topological space $X$ is homeomorphic to $\mathbb QP^\infty$ if and only if $X$ is countable, Hausdorff, and has a countable base $\mathcal B$ of the topology such that for any $n\ge 2$ and basic open sets $U_1,\dots,U_n\in\mathcal B$ the intersection $\bar U_1\cap\dots\cap \bar U_n$ is connected, non-empty, and has zero-dimensional complement $X\setminus (\bar U_1\cap\dots\cap \bar U_n)$.

The proof can be done by a (more-or-less) standard back-and-forth argument.