A fixed-point theorem for definably amenable groups

Uniform Continuity in Definable Groups

“…Thus f (x α ) → f (x). Now (3) ⇒ (1) follows the same proof of the analogous results in [1] or [5]. We refer to the reader to the proof of Theorem 3.5 in [1], although we will mention the changes that we need for the proof to work in the σ-continuous case.…”

“…Now (3) ⇒ (1) follows the same proof of the analogous results in [1] or [5]. We refer to the reader to the proof of Theorem 3.5 in [1], although we will mention the changes that we need for the proof to work in the σ-continuous case. Assume (3) and let • be a σ-continuous action on a non-empty compact set in a locally convex vector space L. So, first we will take F = {U 1 , .…”

“…by condition (2) in the previous paragraph we know that x 0 ∈ U q , while the fact that f (x 0 ) > p implies by (1)…”

“…, n, so that in particular i≤n µ Ui = 1. As the elements in the sets F ∈ D are basic open sets and therefore open sets in the topology of L, the Claims 3.6, 3.7, and 3.8 in [1] remain true. Finally, note that as the σ topology of C is contained in the topology of C, the convergence of the net in the topology of C implies the convergence of the net in the σ-topology.…”