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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)

Citations:

Zbl 0107.02702
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References:

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