After seeing Eric Goubault's work on directed homotopy, where he defines a notion of "locally partially ordered space", I found myself considering the following topos. Does anyone know if it has been used, for instance in algebraic geometry or algebraic topology? What I am trying to capture by the definition is the process of taking the lower semicontinuous reals in [0,1] (classically, [0,1] with its Scott topology, so that the numerical order becomes the specification order) and identifying 0 and 1. Then the specialization from 0 to 1 becomes a non-trivial endomorphism e of 0, so we are forced to consider the modified space as a topos, not a locale. In addition, an extra point will spring into existence, namely the (filtered) colimit of 0 --> 0 --> 0 --> ... e e e The topos is given by a site as follows. Let Q+ be the additive monoid of non-negative rationals and Q/Z the rationals modulo the integers. Q+ acts on Q/Z by addition. Let C be the corresponding category of elements (object = element [q] of Q/Z, morphisms ([q],f): [q] -> [q+f] for f in Q+) and generate a Grothendieck topology on C^op by letting any object [q] of C be cocovered by all the morphisms ([q],f) for f > 0. Let E be the topos corresponding to this site on C^op. Its points are flat presheaves F on C with the condition that if x is in F([q]) then x = ([q],f)y for some f > 0, y in [q+f]. The points arising from the wrapped-round [0,1] are like half-infinite helices (F([q]) = N for all q), and the new point is like an infinite helix (F([q]) = Z for all q). Steve Vickers.