PUB - Publikationen an der Universität Bielefeld

By proving that, if the quotient space S(X) of the connected components ofthe locally compact metric space (X,d) is compact, then the full group I(X,d)of isometries of X is closed in C(X,X) with respect to the pointwise topology,i.e., that I(X,d) coincides in this case with its Ellis' semigroup, we completethe proof of the following: Theorem (a) If S(X) is not compact, I(X,d) need not be locally compact, nor act properly on X. (b) If S(X) is compact, I(X,d) is locally compact but need not act properly on X. (c) If, especially, X is connected, the action (I(X,d),X) is proper.