Let me start by reminding two constructions of topological spaces with such exotic combination of properties:

1) The elements are non-zero integers; base of topology are (infinite) arithmetic progressions with coprime first term and difference.

2) Take $\mathbb{R}^{\infty}\setminus \{0\}$ with product-topology and factorize by the relation $x\sim y \Leftrightarrow x=ty$ for some $t>0$ (infinite-dimensional sphere). Then consider only points with rational coordinates, all but finitely of them vanishing.

The first question is wether are these two examples homeomorphic or somehow related.

The second is an historical one. I've heard that the first example of such space belongs to P. S. Urysohn. What was his example?