It [can be shown](https://mathoverflow.net/questions/46986/countable-connected-hausdorff-space) that the infinite-dimensional rational projective space $\mathbb{QP}^\infty$ is a connected, Hausdorff topological space. What can be said about its homotopy type (is it simply connected, is there any hope to compute its cohomology algebra)?



On the homotopy type of $\mathbb{QP}^\infty$
Any countable Hausdorff space $Q$ is totally path-disconnected. Indeed, if $f:[0,1]\to Q$ is continuous, then its image $X$ is a countable connected compact Hausdorff space. By Urysohn's lemma, then, continuous maps from $X$ to $[0,1]$ separate points. But $X$ is connected, so the image of a continuous map from $X$ to $[0,1]$ is connected, and so must be just a single point since $X$ is countable. Thus $X$ can only have one point, so $f$ is constant.


So, in particular, $\mathbb{Q}\mathbb{P}^\infty$ is totally path-disconnected, and has the weak homotopy type of a countable discrete space.


(In fact, more strongly, any countable $T\_1$ space is totally path-disconnected. See [Why are the integers with the cofinite topology not path-connected?](https://mathoverflow.net/questions/48970/why-are-the-integers-with-the-cofinite-topology-not-path-connected))


