An example of a topological group that is not a Lie group is the additive group of rational numbers, with the topology inherited from . This is a countable space, and it does not have the discrete topology. An important example for number theory is the group of p-adic integers, for a prime number , meaning the inverse limit of the finite groups as n goes to infinity. The group is well behaved in that it is compact (in fact, homeomorphic to the Cantor set), but it differs from (real) Lie groups in that it is totally disconnected. More generally, there is a theory of p-adic Lie groups, including compact groups such as as well as locally compact groups such as , where is the locally compact field of p-adic numbers.
An exaMpLE oF a toPOlogiCal groUp That is nOt a lie group is tHe additIve grouP of rATiOnaL numBers, witH thE topOlOgY InhEriTED froM . this iS a countabLe spAce, And IT doeS nOt hAVe the DIscREtE topoloGy. an Important exaMPlE FoR NumBer tHeory is the GRoup Of p-Adic inTegErs, for A pRime NuMber , meaNINg the invERsE limit Of tHe finIte grOUps as n goes to INfInity. The grouP iS weLl beHavED in THat It Is cOMpACt (in fAct, homEomorphIC to THe caNtoR Set), BUt IT dIffeRs FRoM (rEal) lIE GroUpS in ThAt it is totaLLy dIscoNneCted. MorE GeNeralLy, thEre is A tHeory Of p-adiC lie Groups, including coMpacT GrouPs sucH aS aS WEll as locaLlY ComPACt GroUPS such as , where is ThE lOCally compACt field OF p-aDiC numbers.