question regarding quant-ph/0312044

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\nIn arXiv.org/abs/quant-ph/0312044 (Partiality in Physics,\nBob Coecke and Keye Martin), the Scott topology on a partially\nordered set (P,\\sqsubseteq) is introduced by Definition 2.3:\nU \\subseteq P is open iff\n(i) U is an upper set:\nx \\in U and x \\sqsubseteq y ==&gt; y \\in U\n(usually called the Alexandrov condition)\nand (which is the subject of my question below)\n(ii) U is inaccessible by directed suprema:\nFor every directed S \\subseteq P with supremum \\sqcup S,\n\\sqcup S \\in U ==&gt; S \\cap U \\neq \\emptyset\n(which is apparently usually called the Scott condition)\nI think I intuitively understand the intent of (i), but what\'s\nthe intuitive intent of (ii) wrt the Scott topology and, in particular,\nwrt the authors\' further discussion of "approximation" in Section 2.2?\nThanks,\n--\nJohn Forkosh ( mailto: j@f.com where j=john and f=forkosh )\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>In arXiv.org/abs/http://www.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 )

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