Father-of-two Celyn, who now lives in the north west of England, developed the idea for the film with Pembrokeshireborn Andy Goddard, who

directed Set Fire To The Stars and has a long career in TV, including work on Downton Abbey.

Clearly every (countably) directed set is nonempty, and every countably directed set is directed but not vice versa.

We say that x is way-below y or x approximates y, written x [much less than] y if whenever D is a directed set that has a supremum sup D [greater than or equal to] y, then there is some d [member of] D with x [less than or equal to] d.

Since every countably directed set is directed, we have that x [much less than] y implies [lambda] [[much less than].sub.c] y for all x, y [member of] L.

A [[much less than].sub.c] B if and only if for any countably directed set D [subset or equal to] L, [disjunction] D [member of] [up arrow] B implies D [intersection] [up arrow] A [not equal to] [phi].

Ghosh's erudition and dogged scholarship are extremely impressive; the sense he provides of Weber's craft in shaping, even distorting, the extensive and amorphous array of historical material into a coherent and clearly

directed set of concepts is salutary.

Let [([l.sub.i]).sub.i[member of]I] have a subnet [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] that [theta]-converges to y and J be a directed set. Now suppose that y [member of] X is not a [theta]-cluster point of [([l.sub.i]).sub.i[member of]I].

Since I is a directed set, i [less than or equal to] [i.sub.2] where i [less than or equal to] [i.sub.1] and [i.sub.1] [less than or equal to] [i.sub.2].

This means that J is a directed set. Define k: J [right arrow] I by k(i, A) = i.

This means that (J, [less than or equal to]) is a directed set and partially ordered.

The point is that (whether for the dot product, the parabola, fractions, etc.) finding some

directed set of questions and insights that converges on the discovery of the key mathematical relationship is essential.

A poset in which every

directed set has a supremum is called a dcpo.