The role of connectedness in the structure and the action of group of isometries of locally compact metric spaces

Details The role of connectedness in the structure and the action of group of isometries of locally compact metric spaces The role of connectedness in the structure and the action of group of isometries of locally compact metric spaces

2000-10-09

arxiv.org/abs/math/0010083v1

Mathematics

General Topology

6 pages

Scientific paper

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.

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