I am currently reading the proof of Proposition 2.2. (a direct limit of discrete amenable groups is amenable) in http://people.maths.ox.ac.uk/kar/amenable.pdf . The proof uses the Tychonoff theorem for $[0,1]^{P(G)}$, where $G$ is a discrete group. What is the topology used there on the power set $P(G)$?
2026-02-23 06:00:14.1771826414
Amenability; topology on power sets
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No topology on $\mathcal{P}(G)$ is being used here. Tychonoff's theorem is applied to the product topology on the product $[0,1]^{\mathcal{P}(G)}$ of copies of $[0,1]$ indexed by the set $\mathcal{P}(G)$. The topology on a product space is defined only in terms of products on the factors, not a topology on the index set.