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Letϕ  :   G  →  H{\phi:G\rightarrow H}    be a group homomorphism such that  H  is a totally disconnected locally compact (t.d.l.c.) group and the image of ϕ is dense. We show that all such homomorphisms arise as completions of  G  with respect to uniformities of a particular kind. Moreover,  H  is determined up to a compact normal subgroup by the pair(  G  ,    ϕ   -  1    ⁢   (  L  )    ){(G,\phi^{-1}(L))}   , where  L  is a compact open subgroup of  H . These results generalize the well-known properties of profinite completions to the locally compact setting.

Homomorphisms into totally disconnected, locally compact groups with dense image