By Dana Scott

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N EX}}. :;:;1 DATA TYPES AS LATTICES It is an easy exercise to show that from the definition of "isolated' It follows that a is continuous; and from density, it follows that D is in a one-to-one order preserving correspondence with the fixed points of a. In the last section we introduced an algebra of retracts, much of which carries over to closure operations given the proper definitions. 3 on function spaces, provided we check that the required retracts are closures. 3 (The function space theorem for algebraic lattices).

The countable intersections of open suhsets of Pw are exactly the sets of the form: {xlf(x) = T}, where f: Pw -+ Pw is continuous. -set has this form. Certainly, as we have remarked, every CIl-set has this form. Thus if W is a OJ,. we have: and further, Un = {xIMx) = T), where the frl are suitably chosen continuous functions. mEfn(x)). Clearly g is continuous. and we have: W={xlg(x)=T}, as desired. We tet ii' (lj denote the class of all sets of the form C u. where CEil and U \0 (lj, Similarly fur iI (II,.

1""/. lentitlcs, we return 10 the argument. From what we saw before, given j e. w, there is an n/ such that: Take any two J, j' '= w. We also kno\\' there i" an fl'~ OJ whl'rc: 5fiS Dr\1,\ TYPES AS LA rnCES in vie"! of (7. J <)). But since (f, IT, Il j ) £ (T) U fJi" and U (/, ni') c:: (j, (T/ r; u) U ur, we have: II) U (}', Ill. It follows that and so they are all equal. This determines the fixed nEw we want. Suppose that both! and x are finite sets in Pw. U x). Let A this rime bL: the least equivaicilce relation such that: rlx) Ar(x)Ulf, m) holds for JII III E w.