Uniform universal covers of uniform spaces
The authors develop a generalized covering space theory for a large class of uniform spaces, called coverable spaces, including geodesic metric spaces and connected and locally pathwise connected compact topological spaces, in particular Peano continua and the topological sine curve. The idea is the following : Let \(E\) be an entourage of \(X\). An \(E\)-chain between \(p\) and \(q\) is an ordered set of points \(x_0= p, x_1, \ldots, x_n =q\) such that for all \(i\), \((x_i, x_{i+1})\in E\). Two \(E\)-chains are equivalent if the first one can be obtained from the second one through finitely many steps consisting of adding or removing a point. We denote by \(X_E\) the equivalence classes of \(E\)-chains starting at a fixed point and \(\delta_E\) the group of classes starting and leaving at the fixed point. If \(F\subset E\), we have a natural map \(X_F\to X_E\), and we denote by \(\tilde X = \displaystyle\lim_{\leftarrow} X_E\) and \(\delta_1 = \displaystyle\lim_{\leftarrow} \delta_E\). The uniform covering space \(\tilde X\) of a coverable space (to be defined) is a kind of generalized cover with deck group \(\delta_1\), and with universal and lifting properties that generalize the classical topological situation.
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