In this paper R.H. Bing has constructed his famous example of a countable connected Hausdorff space.

The Bing space $\mathbb B$ is the rational half-plane $\{(x,y)\in\mathbb Q\times \mathbb Q:y\ge 0\}$ endowed with the topology consisting of the sets $U\subset \mathbb B$ such that for any $(a,b)\in U$ there exists $\varepsilon>0$ such that each point $(x,0)$ with $\min\{|x-(a-b/\sqrt{3})|,|x-(a+b/\sqrt{3})|\}<\varepsilon$ belongs to $U$.

It is easy to see that for every rational numbers $a>0$ and $b$ the affine map $f_{a,b}:\mathbb B\to\mathbb B$, $f_{a,b}:(x,y)\mapsto (ax+b,ay)$, is a homeomorphism of the Bing space $\mathbb B$.

This impliees that the action of the homeomorphism group on $\mathbb B$ has at most two orbits.

**Problem.** Is the Bing space $\mathbb B$ topologically homogeneous?

**Remark.** It seems that the first example of a topologically homogeneous countable connected Hausdorff space was constructed by Joseph Martin. A simple transparent example of such space is the rational projective space $\mathbb QP^\infty$ discussed in this MO post. It is interesting who first realized that the space $\mathbb QP^\infty$ is connected?