The role of connectedness in the structure and the action of group of   isometries of locally compact metric spaces
  By proving that, if the quotient space S(X) of the connected components of the 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 complete the 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. 
Antonios Manoussos and Polychronis Strantzalos