In [this paper](http://www.ams.org/journals/proc/1953-004-03/S0002-9939-1953-0060806-9/S0002-9939-1953-0060806-9.pdf) 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](https://pdfs.semanticscholar.org/13e9/fa9bc02a14c9a9b376d3dc3039f3af28d991.pdf). A simple transparent example of such space is the rational projective space $\mathbb QP^\infty$ discussed in this [MO post](https://mathoverflow.net/questions/46986/countable-connected-hausdorff-space/285556#285556). It is interesting who first realized that the space $\mathbb QP^\infty$ is connected?



Is Bing's countable connected space topologically homogeneous?
The Bing space is topologically homogeneous. 
The proof of this fact can be found [here](https://arxiv.org/abs/1712.01964). It is a bit long (to be reproduced here) and uses the standard back-and-forth argument.


