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.