It ha s not been possible to rewrite the entire book for thi s SecondEdition. It would hav e beengratifyingto re survey the th eory of topo-logical groupsin the lightof progressmadein the period 1962-1978, toamplify some sections and cur tai l oth ers, and in generalto profitfromour exp erience s ince thebookwas published.M ark et conditions and oth ercommitments in curr ed by the author s hav e dict at ed oth erwise.We havenon eth elesstaken adv ant ageof the kindnessof Springer-Verlagto makea numb er of improvem ent s in th e t ext and ofcourseto corr ect mi sprint sand math emat ical blunder s.We arein d ebt to the r ead erswho have writt en to us or spoken withus about the text, and we have triedto follow th eir suggestions. We areh appy hereto recordour gratitud e to ROB ERT B.B UR CKEL , W .WI ST ARCOM FORT , RO BERT E .EDW ARD S, ROB ERT E . JAMI SON , JORGE M.L OP Ez,TH EODOR E W. PAUfER, WILLARD A. PARKER ,K ARL R. STROMB ER G, andFR ED THO ELE, as well as to a ho st of otherswho have kindly made sug-gestions to us.Our thank s are due as well to Springer-Verlagfor the ir support ofour work.

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