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Chapter 12
Entourages and (quasi-)uniformities
The techniques of the previous chapters enable us to present with some advan- tage the point-free counterpart of Weil’s approach to not necessarily symmetric uniformities. We are not dealing here with the uniformities based on covers, not because they are less important, but because the covariant technique is not particularly relevant there. The reader interested in such questions can consult, e.g., [57].
12.1. Entourages. Given a locale L, an open sublocale o(E) of the product locale L⊕L is an entourage of L if the diagonal ∆ : L → L⊕L factorizes through the embedding jE : o(E) → L ⊕ L:
∃f xxxxL x|| xxx ∆
jE
12.1.1. Proposition. An open sublocale o(E) of L ⊕ L is an entourage iff
∆∗(E) = 1.
Proof. If jEf = ∆ then ∆∗(E) = f∗jE∗ (E) = f∗(1) = 1. Conversely, if ∆∗(E) =
1 then, for every x ∈ L,
E→∆(x) = ∆(∆∗(E)→x) = ∆(1→x) = ∆(x).
// o(E) L ⊕ L
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