{"id":1484,"date":"2018-05-21T11:14:03","date_gmt":"2018-05-21T09:14:03","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1484"},"modified":"2023-06-18T15:52:04","modified_gmt":"2023-06-18T13:52:04","slug":"dcpos-and-convergence-spaces-i-scott-and-heckmann-convergences","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1484","title":{"rendered":"Dcpos and convergence spaces I: Scott and Heckmann convergences"},"content":{"rendered":"<p>Every dcpo can be seen as a topological space, once we equip it with the Scott topology. And every topological space can be seen as a convergence space, so every dcpo can be seen as a convergence space. <a href=\"https:\/\/www.absint.com\/staff\/rh.htm\">Reinhold Heckmann<\/a> observed that we could see dcpos as convergence spaces in another way [1], with some serendipitous properties. \u00a0We shall see what serendipitous properties\u00a0next time. \u00a0This month, we shall prepare the grounds for that piece of work, by investigating various convergences that can be put on dcpos.<\/p>\n<p>I should say that, as I was writing this, I realized that extensions of those constructions are now being actively researched by <a href=\"https:\/\/www.researchgate.net\/profile\/Hadrian_Andradi\">Hadrian Andradi<\/a> and <a href=\"https:\/\/math.nie.edu.sg\/wkho\/\">Weng Kin Ho<\/a>: see H. Andradi&#8217;s <a href=\"https:\/\/arxiv.org\/search\/?query=andradi&amp;searchtype=all&amp;source=header\">arXiv reports<\/a>. \u00a0I won&#8217;t discuss what they do here, but you may be interested.<\/p>\n<h2>Convergence spaces<\/h2>\n<p>In previous <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=283\">posts<\/a>, I called convergence spaces\u00a0<em>filter spaces<\/em>, following Hyland, but I now prefer to use the term &#8220;convergence space&#8221;, as\u00a0in the standard reference [2]. \u00a0What I used to call convergence spaces are nowadays called limit spaces.<\/p>\n<p>A <em>convergence space<\/em> is a set\u00a0<em>X<\/em> together with\u00a0a relation <em>\u2192<\/em>\u00a0between filters <em>F<\/em>\u00a0of subsets and points <em>x<\/em>, satisfying the following two axioms:<\/p>\n<ul>\n<li>(<em>x<\/em>) <em>\u2192<\/em> x, where (x) is the filter of all subsets of <em>X<\/em> that contain <em>x<\/em> (this is called the <em>principal ultrafilter at<\/em> <em>x<\/em>; the standard notation is <em>x<\/em> with a dot, but I don&#8217;t know how to typeset that in a blog)<\/li>\n<li>If <em>F<\/em> <em>\u2192<\/em> <em>x<\/em> and <em>F&#8217;<\/em> is a filter of subsets that contains <em>F<\/em>, then <em>F&#8217;<\/em> <em>\u2192<\/em> <em>x<\/em>.<\/li>\n<\/ul>\n<p>Alternatively, we may specify\u00a0\u2192 by instead positing an operator lim :\u00a0<strong>F<\/strong><strong>P<\/strong>(<em>X<\/em>)\u00a0\u2192\u00a0<strong>P<\/strong>(<em>X<\/em>) that sends every filter of subsets to the set of its limits. \u00a0(<strong>P<\/strong>(<em>X<\/em>) is the powerset of\u00a0<em>X<\/em>, and\u00a0<strong>F<\/strong><strong>P<\/strong>(<em>X<\/em>) is the set of all filters of subsets of\u00a0<em>X<\/em>. \u00a0Both are ordered by inclusion.) \u00a0We may restate the above axioms as:<\/p>\n<ul>\n<li>x\u00a0\u2208 lim (x), and<\/li>\n<li>lim is monotonic.<\/li>\n<\/ul>\n<p>A\u00a0<em>limit space<\/em> additionally satisfies the following axiom: lim preserves binary intersections. \u00a0A\u00a0<em>pretopological space<\/em> additionally satisfies: lim preserves all intersections.<\/p>\n<p>For every topological space\u00a0<em>X<\/em>, its associated notion of convergence is given by: lim\u00a0<em>F<\/em> is the set of points\u00a0<em>x<\/em> such that <em>N<sub>x<\/sub><\/em>\u00a0is included in in\u00a0<em>F<\/em>, where <em>N<sub>x<\/sub><\/em>\u00a0is the set of neighborhoods of\u00a0<em>x<\/em>. \u00a0This way, every topological space is pretopological\u2014but the converse fails.<\/p>\n<p>Note that, on a topological space\u00a0<em>X<\/em>, lim\u00a0<em>F<\/em> = {<em>x<\/em>\u00a0\u2208\u00a0<em>X<\/em> |\u00a0\u2200\u00a0<em>U<\/em>\u00a0\u2208\u00a0<strong>O<\/strong><em>X<\/em>,\u00a0<em>x<\/em>\u2208<em>U<\/em>\u00a0\u21d2\u00a0<em>U<\/em>\u2208<em>F<\/em>} is the complement of\u00a0{<em>x<\/em>\u00a0\u2208\u00a0<em>X<\/em> | \u2203 <em>U<\/em>\u00a0\u2208\u00a0<strong>O<\/strong><em>X<\/em>,\u00a0<em>x<\/em>\u2208<em>U<\/em>\u00a0and\u00a0<em>U<\/em>\u2209<em>F<\/em>}. \u00a0The latter is the union of all the open sets that fail to be in\u00a0<em>F<\/em>. \u00a0Hence lim\u00a0<em>F<\/em> is the intersection of all the closed sets\u00a0<em>C<\/em> such that\u00a0<em>X<\/em>\u2014<em>C<\/em>\u2209<em>F<\/em>. \u00a0Alternatively, this is the smallest closed set whose complement is not in\u00a0<em>F<\/em>.<\/p>\n<p>Conversely, any notion of convergence gives rise to a notion of open set:\u00a0<em>U<\/em> is said to be\u00a0<em>open<\/em> if and only if for every filter of subsets\u00a0<em>F<\/em>, for every point\u00a0<em>x<\/em> in lim\u00a0<em>F<\/em>, if\u00a0<em>x<\/em> is in\u00a0<em>U<\/em> then\u00a0<em>U<\/em> is in\u00a0<em>F<\/em>. \u00a0In other words, an open set is a set\u00a0<em>U<\/em> that belongs to every filter\u00a0<em>F<\/em> such that lim\u00a0<em>F<\/em> intersects\u00a0<em>U<\/em>. \u00a0The collection of open sets obtained that way is a topology, the so-called\u00a0<em>topological modification<\/em> of the convergence space\u00a0<em>X<\/em>.<\/p>\n<p>Given a\u00a0topology on a set\u00a0<em>X<\/em>, the topological modification of the notion of convergence on\u00a0<em>X<\/em> is the original topology. \u00a0However, given a convergence space, the notion of convergence obtained from its topological modification will not in general be the original convergence: we lose some information by passing to topological modifications.<\/p>\n<h2>Topological convergence on dcpos<\/h2>\n<p>The traditional topology on a dcpo\u00a0<em>X<\/em>, in fact on a poset\u00a0<em>X<\/em> when looked from a domain-theoretical angle, is the Scott topology. \u00a0That determines a notion of convergence that we shall call\u00a0<em>topological\u00a0<\/em><em>convergence<\/em>. \u00a0Let us write lim<sub>T<\/sub> for it.<\/p>\n<p>That is usually defined on nets, see [3], Section II.1. \u00a0On filters, we merely follow the recipe given above for topological spaces:\u00a0for every filter of subsets\u00a0<em>F<\/em>,\u00a0lim<sub>T<\/sub> <em>F<\/em>\u00a0is the smallest Scott-closed set whose complement is not in\u00a0<em>F<\/em>.<\/p>\n<p>There are many other notions of convergence on dcpos, and we shall say that a notion of convergence is\u00a0<em>admissible<\/em>\u00a0if its topological modification is the Scott topology.<\/p>\n<p>In addition to topological convergence lim<sub>T<\/sub>, we willl investigate\u00a0at least Scott convergence and Heckmann convergence.<\/p>\n<h2>Scott convergence<\/h2>\n<p>Scott had introduced a notion of convergence\u00a0on complete lattices, whereby a net (<em>x<sub>i<\/sub><\/em>)<em><sub>i\u2208I, \u2291<\/sub><\/em> converges to\u00a0<em>x<\/em> if and only if\u00a0<em>x<\/em> is below the liminf of the net, sup<em><sub>i\u2208I<\/sub><\/em> inf<em><sub>j\u2292i<\/sub><\/em> <em>x<sub>j<\/sub><\/em>. \u00a0Beware that this is <em>not<\/em> convergence in the Scott topology (but its topological modification\u00a0<em>is<\/em> the Scott topology).<\/p>\n<p>On dcpos, that generalizes to:\u00a0(<em>x<sub>i<\/sub><\/em>)<em><sub>i\u2208I, \u2291<\/sub><\/em> converges to\u00a0<em>x<\/em> if and only if there is a directed family\u00a0<em>D<\/em> whose supremum is above\u00a0<em>x<\/em>, and such that for each element\u00a0<em>d<\/em> of\u00a0<em>D,\u00a0<\/em><em>d<\/em>\u00a0is below <em>x<sub>i<\/sub><\/em>\u00a0for\u00a0<em>i<\/em> large enough. One may instead use ideals instead of directed families, as ideals are exactly the downward closures of directed families.<\/p>\n<p>Weck [5] gave the corresponding definition of Scott convergence in terms of filters. \u00a0Ern\u00e9 generalized that to the case of posets, not just dcpos [4]. \u00a0To make it clear, our notion of Scott convergence is his notion of s<sub>2<\/sub>-convergence, which relies on\u00a0ideals. (The other choices use Frink ideals instead of ideals, and ideals with a supremum instead of all ideals. The latter choice does not make any difference in dcpos.)<\/p>\n<p>Let us call\u00a0<em>Scott convergence<\/em> the resulting notion of convergence. \u00a0That is given as follows: a filter of subsets\u00a0<em>F<\/em> Scott-converges to\u00a0<em>x<\/em> if and only if there is an ideal\u00a0<em>I<\/em> such that\u00a0<em>x<\/em>\u2264sup\u00a0<em>I<\/em> and such that for every element\u00a0<em>z<\/em> of\u00a0<em>I<\/em>,\u00a0\u2191<em>z<\/em> is in\u00a0<em>F<\/em>. \u00a0We write\u00a0lim<sub>S<\/sub> <em>F<\/em> for the set of points that\u00a0<em>F<\/em> Scott-converges to.<\/p>\n<p><strong>Lemma 1.<\/strong> For every filter of subsets\u00a0<em>F<\/em>,\u00a0lim<sub>S<\/sub> <em>F<\/em> \u2286 lim<sub>T<\/sub> <em>F<\/em>.<\/p>\n<p><em>Proof.<\/em>\u00a0Every point <em>x<\/em>\u00a0of\u00a0lim<sub>S<\/sub> <em>F<\/em> is in\u00a0lim<sub>T<\/sub> <em>F<\/em>. \u00a0Indeed, let <em>I<\/em> be an ideal as given above. \u00a0For every Scott-open neighborhood\u00a0<em>U<\/em> of\u00a0<em>x<\/em>,\u00a0<em>I<\/em> intersects\u00a0<em>U<\/em>, say at\u00a0<em>z<\/em>. \u00a0By definition,\u00a0\u2191<em>z<\/em> is in\u00a0<em>F<\/em>. \u00a0Hence the larger set\u00a0<em>U<\/em> is also in\u00a0<em>F<\/em>. \u00a0\u2610<\/p>\n<p><strong>Lemma 2.<\/strong> lim<sub>S\u00a0<\/sub>is admissible.<\/p>\n<p><em>Proof.<\/em> It is a general fact that lim\u00a0\u2286 lim&#8217; implies that all lim&#8217;-open sets are lim-open. \u00a0(Exercise!) \u00a0Using Lemma 1, and the fact that the lim<sub>T<\/sub>-open subsets are the open subsets of the original (Scott) topology, every Scott-open subset is lim<sub>S<\/sub>-open. \u00a0Conversely, let\u00a0<em>U<\/em> be lim<sub>S<\/sub>-open. \u00a0That means that <em>U<\/em> is in\u00a0every filter of subsets <em>F<\/em>\u00a0such that lim<sub>S<\/sub>\u00a0<em>F<\/em> intersects<em> U. \u00a0<\/em>Then:<\/p>\n<ul>\n<li><em>U<\/em> is upwards-closed: if\u00a0<em>x<\/em>\u2264<em>y<\/em> and\u00a0<em>x<\/em> in\u00a0<em>U<\/em>, let\u00a0<em>F<\/em> be the filter of all sets that contain\u00a0<em>y<\/em>. \u00a0One checks that lim<sub>S<\/sub>\u00a0<em>F<\/em>\u00a0=\u00a0\u2193<em>y<\/em>, hence intersects\u00a0<em>U<\/em>. \u00a0So\u00a0<em>U<\/em> is in\u00a0<em>F<\/em>, and that means that\u00a0<em>U<\/em> contains\u00a0<em>y<\/em>.<\/li>\n<li><em>U<\/em> is Scott-open: let <em>D<\/em>=(<em>x<sub>i<\/sub><\/em>)<em><sub>i\u2208I<\/sub><\/em>\u00a0be a directed set, with supremum\u00a0<em>x<\/em> in\u00a0<em>U<\/em>, and assume that no <em>x<sub>i<\/sub><\/em>\u00a0is in\u00a0<em>U<\/em>. \u00a0Let\u00a0<em>F<\/em> be the filter of all sets that contain some <em>x<sub>i<\/sub><\/em>. \u00a0Then\u00a0<em>x<\/em> is in lim<sub>S<\/sub>\u00a0<em>F<\/em>, as one sees by taking\u00a0<em>I<\/em>=\u2193<em>D<\/em>. \u00a0Since\u00a0<em>U<\/em> be lim<sub>S<\/sub>-open,\u00a0<em>U<\/em> is in\u00a0<em>F<\/em>, and that means that some <em>x<sub>i\u00a0<\/sub><\/em>is in\u00a0<em>F<\/em>. \u00a0\u2610<\/li>\n<\/ul>\n<p>The converse of Lemma 1 fails in general, meaning that lim<sub>S<\/sub>\u00a0and\u00a0lim<sub>T<\/sub>\u00a0and in general two different notions of convergence with the same topological modification. \u00a0This is because of\u00a0the following theorem, whose net-theoretical counterpart can be found in [3, Theorem II-1.9]. \u00a0In particular, any non-continuous dcpo will provide a counterexample.<\/p>\n<p><strong>Theorem.<\/strong>\u00a0 For a dcpo\u00a0<em>X<\/em>, lim<sub>S<\/sub>=lim<sub>T<\/sub>\u00a0if and only if <em>X<\/em>\u00a0is continuous.<\/p>\n<p>Proof. Assume\u00a0<em>X<\/em> continuous. \u00a0We show that lim<sub>S<\/sub>=lim<sub>T<\/sub>\u00a0by contradiction. \u00a0Imagine there were a point\u00a0<em>x<\/em>\u00a0of\u00a0<em>X<\/em>,\u00a0and\u00a0a filter of subsets <em>F<\/em>\u00a0such that\u00a0<em>x<\/em> is in lim<sub>T<\/sub>\u00a0<em>F<\/em>\u00a0but not\u00a0in lim<sub>S<\/sub>\u00a0<em>F<\/em>. \u00a0\u00a0For every ideal\u00a0<em>I<\/em> such that\u00a0<em>x<\/em>\u2264sup\u00a0<em>I<\/em>, there is an\u00a0element\u00a0<em>z<\/em> of\u00a0<em>I<\/em>\u00a0such that \u2191<em>z<\/em> is not in\u00a0<em>F<\/em>. \u00a0We consider the ideal\u00a0<em>I<\/em> of all elements way-below\u00a0<em>x<\/em>. \u00a0This is an ideal and\u00a0<em>x<\/em>\u2264sup\u00a0<em>I<\/em> precisely because\u00a0<em>X<\/em> is continuous. \u00a0We obtain that there is a\u00a0<em>z<\/em>\u226a<em>x<\/em> such that\u00a0\u2191<em>z<\/em> is not in\u00a0<em>F<\/em>. Hence the smaller set\u00a0\u219f<em>z<\/em> is not in\u00a0<em>F<\/em> either. \u00a0However,\u00a0\u219f<em>z<\/em> is\u00a0a Scott-open neighborhood of\u00a0<em>x<\/em>, hence must be in\u00a0<em>F<\/em> since\u00a0<em>x<\/em> is in lim<sub>T<\/sub>\u00a0<em>F<\/em>: contradiction.<\/p>\n<p>In the converse direction, assume that lim<sub>S<\/sub>=lim<sub>T<\/sub>. \u00a0Fix a point\u00a0<em>x<\/em> of\u00a0<em>X<\/em>. \u00a0Let <em>F<\/em> be the filter <em>N<sub>x<\/sub><\/em>\u00a0of all\u00a0neighborhoods\u00a0<em>U<\/em> of\u00a0<em>x<\/em>\u00a0in the Scott topology. \u00a0Then\u00a0<em>x<\/em> is in lim<sub>T<\/sub>\u00a0<em>F<\/em>, hence in lim<sub>S<\/sub>\u00a0<em>F<\/em>. \u00a0By definition, there is an ideal\u00a0<em>I<\/em> such that\u00a0<em>x<\/em>\u2264sup\u00a0<em>I<\/em> and\u00a0\u2191<em>z<\/em> is in\u00a0<em>F<\/em>\u00a0for every\u00a0<em>z<\/em> in\u00a0<em>I<\/em>. \u00a0Now \u2191<em>z<\/em> is in\u00a0<em>F<\/em>\u00a0means that\u00a0\u2191<em>z<\/em> is a neighborhood of\u00a0<em>x<\/em>, namely contains a Scott-open neighborhood of <em>x<\/em>; in turn, that\u00a0implies that\u00a0<em>z<\/em> is way-below\u00a0<em>x<\/em>. \u00a0Hence\u00a0<em>I<\/em> is an ideal of elements way-below\u00a0<em>x<\/em>. \u00a0In passing, that implies that sup\u00a0<em>I<\/em>\u2264<em>x<\/em>, hence\u00a0<em>x<\/em>=sup\u00a0<em>I<\/em>. \u00a0Hence\u00a0<em>X<\/em> is continuous. \u00a0\u2610<\/p>\n<p>Note also that lim<sub>S<\/sub>\u00a0is topological (equal to the notion of convergence of its topological modification) if and only if\u00a0lim<sub>S<\/sub>=lim<sub>T<\/sub>, if and only if <em>X<\/em>\u00a0is continuous. \u00a0Hence\u00a0lim<sub>S<\/sub>\u00a0is not topological in general. \u00a0In fact, it is not even a\u00a0<em>limitierung<\/em> in the sense of Weck [5], i.e.,\u00a0<em>X<\/em> with Scott convergence is not even a limit space, unless\u00a0<em>X<\/em> is meet-continuous.<\/p>\n<h2>Heckmann convergence<\/h2>\n<p>Heckmann uses another notion of convergence [1, Definition 18]. \u00a0Here is how it is defined. \u00a0For every subset\u00a0<em>A<\/em> of\u00a0<em>X<\/em>, let us\u00a0write\u00a0<em>A<\/em><sup>\u2193<\/sup>\u00a0for the set of lower bounds of\u00a0<em>A<\/em> (i.e., of points that are below every point of <em>A<\/em>). \u00a0If\u00a0<em>A<\/em>\u2286<em>B<\/em>, then <em>A<\/em><sup>\u2193<\/sup>\u00a0contains <i>B<\/i><sup>\u2193<\/sup>. \u00a0Hence given any filter of subsets\u00a0<em>F<\/em>, the family {<em>A<\/em><sup>\u2193<\/sup> | <em>A<\/em> \u2208\u00a0<em>F<\/em>} of downwards-closed subsets <em>A<\/em><sup>\u2193<\/sup>\u00a0when\u00a0<em>A<\/em> ranges over\u00a0<em>F<\/em> is directed.<\/p>\n<p>Heckmann defines his notion of convergence by:<\/p>\n<p><strong>Definition.<\/strong> lim<sub>H\u00a0<\/sub><em>F<\/em> = cl (\u222a {<em>A<\/em><sup>\u2193<\/sup> | <em>A<\/em> \u2208\u00a0<em>F<\/em>}).<\/p>\n<p>It is interesting to relate this definition to that of Scott convergence. \u00a0As we shall see, the two notions are very close.<\/p>\n<p>To that end, let us define the\u00a0<em>adherence<\/em> adh <em>B<\/em> of a subset\u00a0<em>B<\/em> as the set of points\u00a0<em>x<\/em> such that\u00a0<em>x<\/em>\u2264sup\u00a0<em>D<\/em> for some directed family\u00a0<em>D<\/em> included in\u00a0<em>B<\/em>. \u00a0Clearly, adh\u00a0<em>B<\/em>\u00a0\u2286 cl (<em>B<\/em>). \u00a0In general, the inclusion is proper, and cl (<em>B<\/em>) is obtained by iterating adherences, possibly transfinitely. \u00a0(Formally, cl (<em>B<\/em>) is the least fixed point of the adherence operator containing\u00a0<em>B<\/em>.)<\/p>\n<p>However, in a continuous dcpo,\u00a0adh\u00a0<em>B<\/em>\u00a0= cl (<em>B<\/em>) for every downwards-closed subset <em>B<\/em>. \u00a0Indeed, consider any point\u00a0<em>x<\/em>\u00a0of cl (<em>B<\/em>) outside adh\u00a0<em>B<\/em>, and let\u00a0<em>D<\/em>=\u21a1<em>x<\/em>. \u00a0Since <em>x<\/em> is not in adh\u00a0<em>B<\/em>, and <em>x<\/em>\u2264sup\u00a0<em>D<\/em>,\u00a0<em>D<\/em> cannot be included in\u00a0<em>B<\/em>. \u00a0Let\u00a0<em>y<\/em> in\u00a0<em>D<\/em> (namely,\u00a0<em>y<\/em>\u226a<em>x<\/em>)\u00a0be outside <em>B<\/em>. \u00a0Since\u00a0<em>B<\/em> is downwards-closed,\u00a0\u219f<em>y<\/em> does not intersect\u00a0<em>B<\/em>, and being open, it does not intersect cl (<em>B<\/em>) either: contradiction, since\u00a0<em>x<\/em> is both in\u00a0\u219f<em>y<\/em>\u00a0and in cl (<em>B<\/em>).<\/p>\n<p><strong>Lemma 3.<\/strong> For every filter of subsets\u00a0<em>F<\/em>,\u00a0\u222a {<em>A<\/em><sup>\u2193<\/sup> | <em>A<\/em> \u2208\u00a0<em>F<\/em>} is also the union of all ideals\u00a0<em>I<\/em>\u00a0such that\u00a0for every element\u00a0<em>z<\/em> of\u00a0<em>I<\/em>,\u00a0\u2191<em>z<\/em> is in\u00a0<em>F<\/em>. \u00a0Hence lim<sub>S<\/sub> <em>F\u00a0<\/em>= adh (\u222a {<em>A<\/em><sup>\u2193<\/sup> | <em>A<\/em> \u2208\u00a0<em>F<\/em>}).<\/p>\n<p><em>Proof.<\/em> Let\u00a0<em>x<\/em> belong to some ideal\u00a0<em>I<\/em>\u00a0such that\u00a0for every element\u00a0<em>z<\/em> of\u00a0<em>I<\/em>,\u00a0\u2191<em>z<\/em> is in\u00a0<em>F<\/em>. \u00a0Note that\u00a0(\u2191<em>z<\/em>)<sup>\u2193<\/sup>\u00a0=\u00a0\u2193<em>z<\/em>, so\u00a0<em>x<\/em> is in <em>A<\/em><sup>\u2193<\/sup> where\u00a0<em>A<\/em>=\u2191<em>x<\/em> is in\u00a0<em>F<\/em>. \u00a0Conversely, let\u00a0<em>x<\/em> belong to some\u00a0<em>A<\/em><sup>\u2193<\/sup>\u00a0with\u00a0<em>A<\/em> \u2208\u00a0<em>F<\/em>. \u00a0Let\u00a0<em>I<\/em> be the ideal \u2193<i>x<\/i>. \u00a0For every\u00a0<em>z<\/em> in\u00a0<em>I<\/em>,\u00a0<em>z<\/em> is below\u00a0<em>x<\/em>, which is a lower bound of\u00a0<em>A<\/em>,\u00a0so\u00a0<em>z<\/em> is also a lower bound of\u00a0<em>A<\/em>. \u00a0It follows that\u00a0<em>A<\/em> is included in\u00a0\u2191<i>z<\/i>, so\u00a0\u2191<i>z<\/i>\u00a0is in\u00a0<em>F<\/em>. \u00a0Moreover,\u00a0<em>x<\/em> is clearly in\u00a0<em>I<\/em>. \u00a0That proves the first part of the lemma. \u00a0The second part follows immediately. \u00a0\u2610<\/p>\n<p>We refine Lemma 1 as follows.<\/p>\n<p><strong>Lemma 4.<\/strong> For every filter of subsets\u00a0<em>F<\/em>,\u00a0lim<sub>S<\/sub> <em>F<\/em> \u2286 lim<sub>H<\/sub> <em>F<\/em> \u2286 lim<sub>T<\/sub> <em>F<\/em>.<\/p>\n<p><em>Proof.<\/em>\u00a0lim<sub>S<\/sub> <em>F<\/em> \u2286 lim<sub>H<\/sub> <em>F<\/em> using Lemma 3, since adherence is included in closure. \u00a0We must then show that every point\u00a0<em>x<\/em> of\u00a0lim<sub>H<\/sub> <em>F<\/em>\u00a0is in lim<sub>T<\/sub> <em>F<\/em>. \u00a0Consider an open neighborhood\u00a0<em>U<\/em> of\u00a0<em>x<\/em>. \u00a0Since\u00a0<em>U<\/em> intersects\u00a0lim<sub>H\u00a0<\/sub><em>F<\/em> = cl (\u222a {<em>A<\/em><sup>\u2193<\/sup> | <em>A<\/em> \u2208\u00a0<em>F<\/em>}) (at\u00a0<em>x<\/em>),\u00a0<em>U<\/em> must also intersect\u00a0\u222a {<em>A<\/em><sup>\u2193<\/sup> | <em>A<\/em> \u2208\u00a0<em>F<\/em>}. \u00a0So\u00a0<em>U<\/em> must contain a lower bound\u00a0<em>z<\/em> of some\u00a0<em>A<\/em>\u00a0\u2208\u00a0<em>F<\/em>. \u00a0Then\u00a0<em>U<\/em> must contain\u00a0\u2191<em>z<\/em>, which contains\u00a0<em>A<\/em>. \u00a0It follows that\u00a0<em>U<\/em> is in\u00a0<em>F<\/em>, which shows that\u00a0<em>x<\/em> is in\u00a0lim<sub>T<\/sub> <em>F<\/em>. \u00a0\u2610<\/p>\n<p><strong>Lemma 5.<\/strong> lim<sub>H\u00a0<\/sub>is admissible.<\/p>\n<p>Proof. Recall that lim\u2286lim&#8217; implies that all lim&#8217;-open subsets are lim-open. \u00a0By Lemma 4, the topological modification of lim<sub>H\u00a0<\/sub>is therefore sandwiched between those of lim<sub>T\u00a0<\/sub>and of lim<sub>S<\/sub>\u2014which are both equal to the Scott topology, by Lemma 2. \u00a0\u2610<\/p>\n<p>As a variant of our previous theorem, we have the following which is (of course) due to Heckmann [1, Theorem 25].<\/p>\n<p><strong>Theorem.<\/strong>\u00a0 For a dcpo\u00a0<em>X<\/em>, lim<sub>H<\/sub>=lim<sub>T<\/sub>\u00a0if and only if <em>X<\/em>\u00a0is continuous.<\/p>\n<p><em>Proof.<\/em> If\u00a0<em>X<\/em> continuous, then lim<sub>S<\/sub>=lim<sub>T<\/sub>\u00a0by our previous theorem, hence\u00a0lim<sub>H<\/sub>=lim<sub>T<\/sub>\u00a0by Lemma 4.<\/p>\n<p>In the converse direction, assume that lim<sub>H<\/sub>=lim<sub>T<\/sub>. \u00a0Fix a point\u00a0<em>x<\/em> of\u00a0<em>X<\/em>. \u00a0Let <em>F<\/em> be the filter <em>N<sub>x<\/sub><\/em>\u00a0of all\u00a0neighborhoods\u00a0<em>U<\/em> of\u00a0<em>x<\/em>\u00a0in the Scott topology. \u00a0Then\u00a0<em>x<\/em> is in lim<sub>T<\/sub>\u00a0<em>F<\/em>, hence in lim<sub>H<\/sub>\u00a0<em>F<\/em>. \u00a0For every open neighborhood\u00a0<em>U<\/em> of\u00a0<em>x<\/em>,\u00a0<em>U<\/em> intersects\u00a0lim<sub>H\u00a0<\/sub><em>F<\/em> = cl (\u222a {<em>A<\/em><sup>\u2193<\/sup> | <em>A<\/em> \u2208\u00a0<em>F<\/em>}) (at\u00a0<em>x<\/em>), hence also\u00a0\u222a {<em>A<\/em><sup>\u2193<\/sup> | <em>A<\/em> \u2208\u00a0<em>F<\/em>}. \u00a0By Lemma 3,\u00a0<em>U<\/em> must then intersect some\u00a0ideal\u00a0<em>I<\/em>\u00a0such that\u00a0for every element\u00a0<em>z<\/em> of\u00a0<em>I<\/em>,\u00a0\u2191<em>z<\/em> is in\u00a0<em>F<\/em>. \u00a0In particular, there is a point\u00a0<em>z<\/em> in\u00a0<em>U<\/em> such that\u00a0\u2191<em>z<\/em> is in\u00a0<em>F<\/em>, namely such that\u00a0\u2191<em>z<\/em> is a neighborhood of\u00a0<em>x<\/em>, and, as in the proof of the previous theorem, that implies\u00a0<em>z<\/em>\u226a<em>x<\/em>.<\/p>\n<p>Consider the family <em>D<\/em>\u00a0of points\u00a0<em>z<\/em> such that\u00a0\u2191<em>z<\/em> is a neighborhood of\u00a0<em>x<\/em>. \u00a0That is non-empty, by the argument we have just seen with\u00a0<em>U<\/em>=<em>X<\/em>. \u00a0<em>D<\/em> is directed: for all\u00a0<em>z&#8217;<\/em>,\u00a0<em>z&#8221;<\/em> in\u00a0<em>D<\/em>, the interiors int (\u2191<em>z&#8217;<\/em>) and\u00a0int (\u2191<em>z&#8221;<\/em>) both contain\u00a0<em>x<\/em>, so\u00a0<em>U<\/em>=int (\u2191<em>z&#8217;<\/em>) \u2229 int (\u2191<em>z&#8221;<\/em>) is an open neighborhood of\u00a0<em>x<\/em>; applying the argument we have just seen, there is a point\u00a0<em>z<\/em> in\u00a0<em>U<\/em> such that\u00a0\u2191<em>z<\/em> is a neighborhood of\u00a0<em>x<\/em>, namely such that\u00a0<em>z<\/em> is in\u00a0<em>D<\/em>. \u00a0We also know that every element of\u00a0<em>D<\/em> is way-below\u00a0<em>x<\/em>. \u00a0Since the intersection of all open neighborhoods of\u00a0<em>x<\/em> is\u00a0\u2191<em>x<\/em>, and each contains some\u00a0\u2191<em>z<\/em> with\u00a0<em>z<\/em> in\u00a0<em>D<\/em>, the intersection of all sets\u00a0\u2191<em>z<\/em> with\u00a0<em>z<\/em> in\u00a0<em>D<\/em> is included in\u00a0\u2191<i>x<\/i>, hence equal to it. \u00a0It follows that\u00a0<em>x<\/em>=sup\u00a0<em>D<\/em>. \u00a0This exhibits\u00a0<em>x<\/em> as the supremum of a directed family of elements way-below\u00a0<em>x<\/em>, so\u00a0<em>X<\/em> is continuous. \u00a0\u2610<\/p>\n<p>As for\u00a0lim<sub>S<\/sub>, lim<sub>H\u00a0<\/sub>is topological if and only if lim<sub>H<\/sub>=lim<sub>T<\/sub>, if and only if <em>X<\/em>\u00a0is continuous. \u00a0Hence\u00a0lim<sub>H<\/sub>\u00a0is not topological in general.<\/p>\n<ol>\n<li>Reinhold Heckmann. <a href=\"https:\/\/pdfs.semanticscholar.org\/e257\/3a1b8523c59087f58a7c4031f5f97a4a6e40.pdf\">A Non-Topological View of Dcpos as Convergence Spaces<\/a>.<br \/><a class=\"publication-title-link\" title=\"Go to Theoretical Computer Science on ScienceDirect\" href=\"https:\/\/www.sciencedirect.com\/science\/journal\/03043975\">Theoretical Computer Science<\/a><a title=\"Go to table of contents for this volume\/issue\" href=\"https:\/\/www.sciencedirect.com\/science\/journal\/03043975\/305\/1\">Volume 305, Issues 1\u20133<\/a>, 18 August 2003, Pages 159-186.<\/li>\n<li>Szymon Dolecki and Fr\u00e9d\u00e9ric Mynard. \u00a0<a href=\"https:\/\/www.worldscientific.com\/worldscibooks\/10.1142\/9012\">Convergence foundations of topology<\/a>. \u00a0World scientific, July 2016, 568 pages.<\/li>\n<li>Gerhard Gierz, Karl Heinrich Hofmann, Klaus Keimel, Jimmie D. Lawson, Michael W. Mislove, and Dana S. Scott. Continuous Lattices and Domains. Number 93 in Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2003.<\/li>\n<li>Marcel Ern\u00e9. Scott convergence and Scott topology in partially ordered sets ii. In Continuous Lattices, volume 871, pages 61\u201396. Springer-Verlag Berlin, Heidelberg, New York, 1981.<\/li>\n<li>S. Weck. Scott convergence and Scott topology in partially ordered sets I, in: B. Banaschewski, R.-E. Homann (Eds.), Continuous Lattices, Proc. Conf. Bremen, 1979, Lecture Notes in Mathematics, volume 871, Springer, Berlin, 1981, pages 372\u2013383.<\/li>\n<\/ol>\n<p style=\"text-align: right;\">\u2014 <a href=\"https:\/\/www.lsv.ens-paris-saclay.fr\/~goubault\/?l=en\" rel=\"attachment wp-att-993\">Jean Goubault-Larrecq<\/a> (May 21st, 2018)<img loading=\"lazy\" decoding=\"async\" class=\"wp-image-993 alignright\" src=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/wp-content\/uploads\/2016\/08\/jgl-2011.png\" alt=\"jgl-2011\" width=\"32\" height=\"44\" \/><\/p>\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Every dcpo can be seen as a topological space, once we equip it with the Scott topology. And every topological space can be seen as a convergence space, so every dcpo can be seen as a convergence space. Reinhold Heckmann &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1484\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-1484","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1484","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1484"}],"version-history":[{"count":16,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1484\/revisions"}],"predecessor-version":[{"id":6894,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1484\/revisions\/6894"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1484"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}