Jeremy Irons Dragons" (2000) and Rupert Gould in "Longitude" (2000). He played the Über-Morlock in the film "The Time Machine" (2002). In 2004, Irons played Severus Snape in Comic Relief's "Harry Potter" parody, "Harry Potter and the Secret Chamberpot of Azerbaijan". In 2005, he appeared in the films "Casanova" opposite Heath Ledger, and Ridley Scott's "Kingdom of Heaven". He has co-starred with John Malkovich in two films, "The Man in the Iron Mask" (1998) and "Eragon" (2006), though they did not have any scenes together in the latter. In 2008, Irons co-starred with Ed Harris and Viggo Mortensen in "Appaloosa", directed by
Directed set While the definition of a directed set is for an "upward-directed" set (every pair of elements has an upper bound), it is also possible to define a downward-directed set in which every pair of elements has a common lower bound. A subset of a poset is downward-directed if and only if its upper closure is a filter. Directed subsets are used in domain theory, which studies directed complete partial orders. These are posets in which every upward-directed set is required to have a least upper bound. In this context, directed subsets again provide a generalization of convergent sequences. Directed set