question regarding quant-ph/0312044

From: John Forkosh (john_at_SeeSigForAddress.invalid.com)
Date: 07/19/04 

Date: 19 Jul 2004 04:09:29 -0400

In arXiv.org/abs/quant-ph/0312044 (Partiality in Physics,
Bob Coecke and Keye Martin), the Scott topology on a partially
ordered set (P,\sqsubseteq) is introduced by Definition 2.3:
  U \subseteq P is open iff
     (i) U is an upper set:
             x \in U and x \sqsubseteq y ==> y \in U
          (usually called the Alexandrov condition)
  and (which is the subject of my question below)
     (ii) U is inaccessible by directed suprema:
             For every directed S \subseteq P with supremum \sqcup S,
             \sqcup S \in U ==> S \cap U \neq \emptyset
          (which is apparently usually called the Scott condition)
I think I intuitively understand the intent of (i), but what's
the intuitive intent of (ii) wrt the Scott topology and, in particular,
wrt the authors' further discussion of "approximation" in Section 2.2?
Thanks, 

-- 
John Forkosh  ( mailto:  j@f.com  where j=john and f=forkosh )