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Every locally compact group $G$ contains an open, closed, almost connected subgroup. This is a variant Dantzig's theorem. Pull back an open, compact subgroup in the totally disconnected group $G/G_0$ along the surjection $G \twoheadrightarrow G/G_0$, where $G_0$ is the connected component of the identity.

An almost connected, locally compact group $G'$ admits a net of normal compact subgroups $\cap N = \{ 1 \}$ and such that $G'/N$ is isomorphic to a Lie group, i.e. $G'$ is a projective limit of Lie groups:

$$ G' = \lim\limits_{\leftarrow} G'/N.$$

This is the solution to Hilbert's 5th problem.

Now, one is able to define everything via inductive and projective limits.

Edit: As far as I understand, these are the systems of proejctive Lie groups also considered in the refernce of Igor Rivin. Terrence Tao has written something about this on his blog: http://terrytao.wordpress.com/2011/10/08/254a-notes-5-the-structure-of-locally-compact-groups-and-hilberts-fifth-problem/ One personal note: You do not necessarily need to understand the proofs, to apply the theorems.

Every locally compact group $G$ contains an open, closed, almost connected subgroup. This is a variant of van Dantzig's theorem. Pull back an open, compact subgroup in the totally disconnected group $G/G_0$ along the surjection $G \twoheadrightarrow G/G_0$, where $G_0$ is the connected component of the identity.

An almost connected, locally compact group $G'$ admits a net of normal compact subgroups $\cap N = \{ 1 \}$ and such that $G'/N$ is isomorphic to a Lie group, i.e. $G'$ is a projective limit of Lie groups:

$$ G' = \lim\limits_{\leftarrow} G'/N.$$

This is the solution to Hilbert's 5th problem.

Now, one is able to define everything via inductive and projective limits.

Edit: As far as I understand, these are the systems of proejctive Lie groups also considered in the refernce of Igor Rivin. Terrence Tao has written something about this on his blog: http://terrytao.wordpress.com/2011/10/08/254a-notes-5-the-structure-of-locally-compact-groups-and-hilberts-fifth-problem/ One personal note: You do not necessarily need to understand the proofs, to apply the theorems.

Every locally compact group $G$ contains an open, closed, almost connected subgroup. This is a variant Dantzig's theorem. Pull back an open, compact subgroup in the totally disconnected group $G/G_0$ along the surjection $G \twoheadrightarrow G/G_0$, where $G_0$ is the connected component of the identity.

An almost connected, locally compact group $G'$ admits a net of normal compact subgroups $\cap N = \{ 1 \}$ and such that $G'/N$ is isomorphic to a Lie group, i.e. $G'$ is a projective limit of Lie groups:

$$ G' = \lim\limits_{\leftarrow} G'/N.$$

This is the solution to Hilbert's 5th problem.

Now, probably one is able to define everything via inductive and projective limits.

Every locally compact group $G$ contains an open, closed, almost connected subgroup. This is a variant Dantzig's theorem. Pull back an open, compact subgroup in the totally disconnected group $G/G_0$ along the surjection $G \twoheadrightarrow G/G_0$, where $G_0$ is the connected component of the identity.

An almost connected, locally compact group $G'$ admits a net of normal compact subgroups $\cap N = \{ 1 \}$ and such that $G'/N$ is isomorphic to a Lie group, i.e. $G'$ is a projective limit of Lie groups:

$$ G' = \lim\limits_{\leftarrow} G'/N.$$

This is the solution to Hilbert's 5th problem.

Now, one is able to define everything via inductive and projective limits.

Edit: As far as I understand, these are the systems of proejctive Lie groups also considered in the refernce of Igor Rivin. Terrence Tao has written something about this on his blog: http://terrytao.wordpress.com/2011/10/08/254a-notes-5-the-structure-of-locally-compact-groups-and-hilberts-fifth-problem/ One personal note: You do not necessarily need to understand the proofs, to apply the theorems.

Every locally compact group $G$ contains an open, closed, almost connected subgroup. This originates from Dantzig's theorem applied to the map and pulling back an open, compact subgroup in the totally disconnected group $G/G_0$ along the surjection $G \twoheadrightarrow G/G_0$, where $G_0$ is the connected component of the identity.

An almost connected, locally compact group $G'$ admits a net of normal compact subgroups $\cap N = \{ 1 \}$ and such that $G'/N$ is isomorphic to a Lie group, i.e. $G'$ is a projective limit of Lie groups:

$$ G' = \lim\limits_{\leftarrow} G'/N.$$

This is the solution to Hilbert's 5th problem.

Now, probably one is able to define everything via inductive and projective limits.

Every locally compact group $G$ contains an open, closed, almost connected subgroup. This is a variant Dantzig's theorem. Pull back an open, compact subgroup in the totally disconnected group $G/G_0$ along the surjection $G \twoheadrightarrow G/G_0$, where $G_0$ is the connected component of the identity.

An almost connected, locally compact group $G'$ admits a net of normal compact subgroups $\cap N = \{ 1 \}$ and such that $G'/N$ is isomorphic to a Lie group, i.e. $G'$ is a projective limit of Lie groups:

$$ G' = \lim\limits_{\leftarrow} G'/N.$$

This is the solution to Hilbert's 5th problem.

Now, probably one is able to define everything via inductive and projective limits.