All topologies come from quasi-uniformities
by
Isidore Fleischer
C.R.M., Univ. de Montreal

This is an old (1960) result of Csaszar. It will entail that also all topologies come from continuity spaces once these have been shown quasi-uniformizable. A continuity function is a map d from X×X to a down-directed additive semigroup P which satisfies the triangle inequality; P is in addition equipped with a selfmap r→ r/2 and d(x, x) = 0 identically. (The other axioms are irrelevant for our purpose). A quasi-uniformity (on set X) is a filter on X×X refined by its filter of composites U○V (xU○Vz iff xUy and yVz for some y) all of whose elements contain the diagonal. The elements of P model a filterbase: indeed, the inverse images of the "intervals" [0, p] are a filterbase for the quasi-uniformity. Hence assign each p the set of { r ≤ p }. To see that composites are preserved, observe that the filter corresponding to (r∩p)/2 refines both filter elements corresponding to r and p.

R. Kopperman, All topologies come from generalized metrics, Amer. Math. Monthly 95 (1988), no. 2, 89-97.

Date received: February 27, 2008