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Hilbert’s fifth problem for local groups. (English) Zbl 1219.22004
The author has started two versions of Hilbert’s fifth problem written below after some definitions:
(a) A local group is a tuple \((G,1,\iota,t, p)\) (or simply, \(G\) in short) where \(G\) is a Hausdorff topological space with a distinguished element \(1\in G\), an \(\iota:\Lambda\to G\) (the inversion map) and \(p:\Omega\to G\) (the product map) are continuous functions with \(\Lambda\subseteq G\) and open \(\Omega\subseteq G\times G\), such that \(1\in\Lambda\), \(\{1\}\times G\subseteq\Omega\), \(G\times \{1\}\subseteq\Omega\), and for all \(x,y,z\in G\):
(1) \(p(1, x)= p(x, 1)= x\); (2) if \(x\in\Lambda\), then \((x, \iota(x))\in\Omega\), \((\iota(x), x)\in\Omega\) and \(p(x,\iota(x))= p(\iota(x), x)= 1\); (3) if \((x, y), (y, z)\in\Omega\) and \((p(x, y),z)\), \((x, p(y,z))\in\Omega\), then (A) \(p(p(x,y),z)= p(x,p(y,z))\);
(b) \(G\) is locally Euclidean if there is an open neighbourhood of 1 homeomorphic to an open subset of \(\mathbb{R}^n\) for some \(n\);
(c) \(G\) is a local Lie group if \(G\) admits a \(C^\infty\) structure such that the maps \(\iota\) and \(p\) are \(C^w\);
(d) with \(U\) an open neighbourhood of 1 in \(G\), the restriction of \(G\) to \(U\) is the local group \(G/U:= (U,\iota/\Lambda_U, p/\Omega_U)\), where \(\Lambda_U:= \Lambda\cap U\cap \iota^{-1}(U)\) and \(\Omega_U:= (U\times U)\cap p^{-1}(U)\);
(e) \(G\) is globalizable if there is a topological group \(H\) and an open neighbourhood \(U\) of \(1_H\) in \(H\) such that \(G= H/U\).
Local \(H5\) first form: If \(G\) is a locally Euclidean local group, then some restriction of \(G\) is a local Lie group. Local \(H5\) second form: If \(G\) is a locally Euclidean local group, then some restriction of \(G\) is globalizable.
The equivalence of the two forms is established. The author follows the non-standard treatment of Hilbert’s fifth problem given by Hirschfeld; in dealing with local \(H5\), the author has a simplified way in the non-standard method.

22A30 Other topological algebraic systems and their representations
22E99 Lie groups
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