## A conjecture of Gleason on the foundations of geometry.(English)Zbl 1286.54034

In [Algebr. Topol. Foundations Geom., Proc. Colloq. Utrecht, August 1959, 39–44 (1962; Zbl 0107.02702)], A. M. Gleason posed a conjecture on the possibility of topologizing a group acting on a topological space so that a continuous action would ensue. The conjectured conditions on the group $$G$$ and the space $$M$$ were: (1) $$M$$ is a Polish space and $$G$$ is a group of homeomorphisms that acts transitively on $$M$$, and (2) there are $$n\in \mathbb N$$ and $$\langle m_1,\dots,m_n\rangle\in M^n$$ such that $$g\mapsto\langle gm_1,\dots,gm_n\rangle$$ is injective and for all $$m\in M$$ the subset $$\{\langle gm_1,\dots,gm_n,gm\rangle:g\in G\}$$ of $$M^{n+1}$$ is analytic.
The authors disprove the conjecture but show that some modifications of (2) do lead to positive results: one should require that $$\{\langle gm_1,\dots,gm_n\rangle:g\in G\}$$ is a $$G_\delta$$-set and that $$\{\langle gm_1,\dots,gm_n,gm\rangle:g\in G\}$$ is Borel. In this case one can even take a point in $$M^{\mathbb N}$$ rather than in a finite power. Examples show that the conditions are not necessary.
Reviewer: K. P. Hart (Delft)

### MSC:

 54H15 Transformation groups and semigroups (topological aspects) 54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)

### Keywords:

topological transformation group; Lie group; manifold

Zbl 0107.02702
Full Text:

### References:

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