Topological automorphism groups of chains (Q2735240)

Let \(X\) be a chain and let \(A(X)\) denote the automorphism group of \(X\). The author shows that any set-open topology on \(A(X)\) coincides with the pointwise topology and that \(A(X)\) is a topological lattice-ordered group with respect to this topology. \(A(X)\) is a function space. The topological properties of connectedness and compactness in \(A(X)\) are investigated. \(A(X)\) is connected (resp. totally disconnected) if and only if \(X\) is connected (resp. totally disconnected). The automorphism group of a doubly homogeneous chain is generated by any neighbourhood of the identity. The author also establishes three criteria for a subset to be compact in \(A(X)\) and he uses the described techniques to develop an approach to robustness of aggregation procedures.