Quotients of uniform spaces (Q2502965)

The author states that ``to give up on quotients of uniform spaces simply because such quotients do not always have a compatible uniform structure is a bit like giving up on quotient groups because the quotient via an arbitrary subgroup may not have a compatible group structure.'' His paper is an interesting, important defense of this thesis. The construct in his paper that plays the role in the theory of uniform spaces of a normal subgroup in the theory of groups is the notion of an equiuniform group of bijections of a uniform space. A group \(G\) of bijections of a uniform space \(X\) is equiuniform provided that for each entourage \(U\) there is an entourage \(V\) such that for each \(g\in G\), \(g\times g(V)\subseteq U\). The set of all orbits is denoted by \(X/G\) and the quotient map taking each \(x\in X\) to its orbit is denoted by \(\pi\). Equiuniform groups abound. In a metric space any group of isometrics is equiuniform, and the other pier of uniform spaces, a topological group, is dealt with in Proposition 24. If \(H\) is a subgroup of a topological group \(G\) with its right uniformity, then \(H\) is equiuniform when considered as a group of right translations and is equiuniform when considered as a group of left translations exactly when \(H\) its weakly central. In either case, if \(G/H\) is endowed with the same (i.e. left or right) uniformity as G, then the topology on \(G/H\) is the quotient topology and the quotient map \(\pi:G\to G/H\) is identical to the usual quotient map of topological groups. Theorem 11 is a key result. Let \(G\) be an equiuniform group of bijections of a uniform space \(X\). Then \({\mathcal U}:= \{\pi*\pi(E)\): \(E\) is an entourage in \(X\}\) is the unique uniformity on the quotient \(X/G\) such that the quotient map \(\pi\) is bi-uniformly continuous. Moreover this map is an open mapping and the topology of \(U\) is the quotient topology. The paper gives numerous other allied results, some dealing with groups of homeomorphisms and some dealing with conditions more stringent than being equiuniform. The last section deals with inverse limits of group actions on uniform spaces. Reviewer's remarks: Some of the results of this paper are so fundamental that anyone giving a course covering uniform spaces will want to incorporate them. Fortunately, but not fortuitously, the paper is well suited to this purpose, for the author has obviously worked hard to make his paper widely accessible.