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?

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    $\begingroup$ Concerning this question: "who first realized that the space $\mathbb{Q}P^\infty$ is connected?" Well, I realized it myself after meditating on the properties of $\mathbb{R}P^\infty$ mentioned in the paper by Gelfand and Fuks (English: link.springer.com/article/10.1007%2FBF01076008 Russian: mi.mathnet.ru/faa2842 see p.3 in introduction) My attention to this question (they do not prove their claim in the cited paper) was attracted by Nikolai Ivanov. Of course, most probably I am not the first and it was noted much before. $\endgroup$ – Fedor Petrov Nov 21 '17 at 18:21

The Bing space is topologically homogeneous. The proof of this fact can be found here. It is a bit long (to be reproduced here) and uses the standard back-and-forth argument.

  • $\begingroup$ Are $\mathbb B$ and $\mathbb Q P^\infty$ homeomorphic? $\endgroup$ – André Henriques Dec 20 '17 at 22:32
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    $\begingroup$ @AndréHenriques $\mathbb B$ and $\mathbb Q P^\infty$ are not homeomorphic. They can be distinguished by the following properties: (1) the intersection $\bar U_1\cap\dots\cap \bar U_n$ of the closures of any non-empty open sets $U_1,\dots,U_n$ in $\mathbb Q P^\infty$ is not empty; (2) the Bing space contains three non-empty open sets $U_1,U_2,U_2$ such that $\bar U_1\cap \bar U_2\cap\bar U_3$ is empty. $\endgroup$ – Taras Banakh Dec 21 '17 at 6:17

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