{"id":1184,"date":"2017-04-28T23:02:15","date_gmt":"2017-04-28T21:02:15","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1184"},"modified":"2022-11-19T15:21:43","modified_gmt":"2022-11-19T14:21:43","slug":"bounded-complete-and-dcpo-models-of-t1-spaces","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1184","title":{"rendered":"Bounded complete and dcpo models of T1 spaces"},"content":{"rendered":"<p><a href=\"https:\/\/math.nie.edu.sg\/dszhao\/\">Dongsheng Zhao<\/a> recently mentioned one recent result of his and Xiaoyong Xi to me, on models of spaces [1], which leads to a curious situation.<\/p>\n<p>Let us call <em>model<\/em> of a topological space <em>X<\/em> any poset <em>Y<\/em> whose subset of Max <em>Y<\/em> of maximal points, with the subspace topology from the Scott topology on <em>Y<\/em>, is homeomorphic to <em>X<\/em>.\u00a0 This extends the definition in the <a href=\"https:\/\/www.cambridge.org\/gb\/knowledge\/isbn\/item7069109\/Non-Hausdorff%20Topology%20and%20Domain%20Theory\/?site_locale=en_GB\">book<\/a> (Proposition 7.7.16), where I required <em>Y <\/em>to be a dcpo.<\/p>\n<p>Clearly, any space that has a model must be T<sub>1<\/sub>.\u00a0 But which T<sub>1<\/sub> spaces exactly do have models?\u00a0 And which do have models of a particular kind, for example a model that is a dcpo, or a model that is a continuous dcpo, or etc.?<\/p>\n<p>In the <a href=\"https:\/\/www.cambridge.org\/gb\/knowledge\/isbn\/item7069109\/Non-Hausdorff%20Topology%20and%20Domain%20Theory\/?site_locale=en_GB\">book<\/a>, I mention two theorems by Keye Martin, extending previous results by Jimmie Lawson: the metrizable spaces that have a continuous dcpo model are exactly the completely metrizable spaces (Theorem 7.7.21), and the T<sub>3<\/sub> spaces that have an \u03c9-continuous model are exactly the Polish spaces.<\/p>\n<p>In these pages, I have already mentioned the following theorem by <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=70\">Mummert and Stephan<\/a> [2]: the countably-based T<sub>1<\/sub> spaces that have an \u03c9-continuous model Y are those that are Choquet-complete.\u00a0 And there are many more results of that kind.<\/p>\n<p>The curious situation I would like to describe today is the following:<\/p>\n<ul>\n<li>Every T<sub>1<\/sub> space has a bounded complete poset model.<\/li>\n<li>Every T<sub>1<\/sub> space has a dcpo model.<\/li>\n<li>Some T<sub>1<\/sub> spaces have <em>no<\/em> bounded complete dcpo model, i.e., you cannot have both &#8220;bounded complete&#8221; and &#8220;dcpo&#8221; at the same time.<\/li>\n<\/ul>\n<p>I will develop each point in turn.<\/p>\n<h2>Every T<sub>1<\/sub> space has a bounded complete poset model<\/h2>\n<p>This is due to Zhao [3] and Ern\u00e9 [4] independently.\u00a0 We can in fact require the bounded complete poset model to be algebraic.<\/p>\n<p>If <em>X<\/em> were not just T<sub>1<\/sub> but also sober and locally compact, then we could define a model <em>Y<\/em> as <strong>Q<\/strong>(<em>X<\/em>), its Smyth powerdomain, consisting of the non-empty compact saturated subsets of <em>X<\/em> under reverse inclusion (Corollary 8.3.27 in the <a href=\"https:\/\/www.cambridge.org\/gb\/knowledge\/isbn\/item7069109\/Non-Hausdorff%20Topology%20and%20Domain%20Theory\/?site_locale=en_GB\">book<\/a>).\u00a0 Now note that by the Hofmann-Mislove theorem (Theorem 8.3.2), <strong>Q<\/strong>(<em>X<\/em>) is isomorphic to the poset of proper Scott-open filters of open subsets of <em>X<\/em>, when <em>X<\/em> is sober.\u00a0 (A filter is proper, or non-trivial, if it does not contain the empty set.)\u00a0 We may hope of getting rid of the sobriety assumption by moving from <strong>Q<\/strong>(<em>X<\/em>) to proper Scott-open filters.<\/p>\n<p>Zhao&#8217;s move to avoid local compactness is to consider the poset of proper, not necessarily Scott-continuous, filters of open subsets of <em>X<\/em>.\u00a0 That is fine, but if you do so, and equate the elements <em>x<\/em> of\u00a0<em>X<\/em> with the filters of open neighborhoods of\u00a0<em>x<\/em>, then those will not be maximal.\u00a0 Zhao additionally restricts to those proper filters <em>F<\/em> such that the intersection of all the elements of <em>F<\/em> is non-empty.<\/p>\n<p>The poset <strong>F<\/strong>(<strong>O<\/strong>(<em>X<\/em>)) of proper filters of open subsets of <em>X<\/em> is not something new.\u00a0 It is the subject of Exercise 9.3.10 of the <a href=\"https:\/\/www.cambridge.org\/gb\/knowledge\/isbn\/item7069109\/Non-Hausdorff%20Topology%20and%20Domain%20Theory\/?site_locale=en_GB\">book<\/a>: <strong>F<\/strong>(<strong>O<\/strong>(<em>X<\/em>)) is a Scott domain, that is, an algebraic bc-domain, directed suprema are just unions, and the finite elements of <strong>F<\/strong>(<strong>O<\/strong>(<em>X<\/em>)) are the filters \u25a0<em>U<\/em>, <em>U<\/em> open in <em>X<\/em>, where \u25a0<em>U<\/em> is the filter of all open supersets of <em>U<\/em>.<\/p>\n<p>Zhao&#8217;s poset, call it <strong>Zh<\/strong>(<em>X<\/em>), is the sub-poset of those filters <em>F<\/em> such that there is a point of <em>X<\/em> that lies in every element of <em>F<\/em>.\u00a0 (This is a slight reformulation of what I announced earlier.)\u00a0 <strong>Zh<\/strong>(<em>X<\/em>), contrarily to <strong>F<\/strong>(<strong>O<\/strong>(<em>X<\/em>)), need not be a dcpo.\u00a0 However, if a family of filters (<em>F<sub>i<\/sub><\/em>)<em><sub>i \u2208 I<\/sub><\/em> of <strong>Zh<\/strong>(<em>X<\/em>) has a supremum <em>F<\/em> in <strong>Zh<\/strong>(<em>X<\/em>), then it must be the union \u222a<em><sub>i <\/sub><\/em><em>F<sub>i<\/sub><\/em>. Indeed, there must be a point <em>x<\/em> that is in every element of <em>F<\/em>, hence in every element of every <em>F<sub>i<\/sub><\/em>; that shows that \u2229<em><sub>i <\/sub><\/em><em>F<sub>i<\/sub><\/em> is in <strong>Zh<\/strong>(<em>X<\/em>), and is the required supremum.\u00a0 In turn, the fact that directed suprema in <strong>Zh<\/strong>(<em>X<\/em>) are computed as in the ambient dcpo <strong>F<\/strong>(<strong>O<\/strong>(<em>X<\/em>)) shows that the filters \u25a0<em>U<\/em>, where <em>U<\/em> is non-empty and open, are finite in <strong>Zh<\/strong>(<em>X<\/em>), leading us to show that <strong>Zh<\/strong>(<em>X<\/em>) is algebraic.\u00a0 Finally, it is easy to show that <strong>Zh<\/strong>(<em>X<\/em>) is bounded complete: if <em>F<\/em> and <em>G<\/em> are two filters in <strong>Zh<\/strong>(<em>X<\/em>) with an upper bound, then their least upper bound in <strong>F<\/strong>(<strong>O<\/strong>(<em>X<\/em>)), which happens to be the set of open sets <em>W<\/em> that contains the intersection of an element of <em>F<\/em> and of an element of <em>G<\/em>, is also in <strong>Zh<\/strong>(<em>X<\/em>): as with the computation of directed suprema, take a point <em>x<\/em> that is in every element of the chosen upper bound, and realize that it must be in every <em>W<\/em> constructed as above.<\/p>\n<p>If <em>X<\/em> is T<sub>0<\/sub>, then the space <em>X<\/em> embeds in <strong>Zh<\/strong>(<em>X<\/em>) through the familiar map \u03b7 that maps every point<em> x<\/em> to its set of open neighborhoods.\u00a0 That is continuous because \u03b7<sup>-1<\/sup>(\u2191\u25a0<em>U<\/em>) is just <em>U<\/em>.\u00a0 That also shows that \u03b7 is almost open, and it is injective since <em>X<\/em> is T<sub>0<\/sub>.<\/p>\n<p>If <em>X<\/em> is T<sub>1<\/sub>, then Max <strong>Zh<\/strong>(<em>X<\/em>) consists exactly of the filters of the form \u03b7(<em>x<\/em>), <em>x<\/em> in <em>X<\/em>.\u00a0 Indeed, if <em>F<\/em> is a maximal element of <strong>Zh<\/strong>(<em>X<\/em>), and <em>x<\/em> is one of the points that belongs to every element of <em>F<\/em>, then, first, <em>F<\/em> must contain all the open neighborhoods of <em>x<\/em>.\u00a0 Otherwise, we could add all the intersections of open neighborhoods of <em>x<\/em> with elements of <em>F<\/em> to <em>F<\/em> and still obtain an element of <strong>Zh<\/strong>(<em>X<\/em>).\u00a0 Then there cannot be two distinct points <em>x<\/em> and <em>y<\/em> that both belong to every element of <em>F<\/em>: since<em> X <\/em>is<em> T<sub>1<\/sub><\/em>, take an open neighborhood <em>U<\/em> of <em>x<\/em> that does not contain <em>y<\/em>, and an open neighborhood <em>V<\/em> of <em>y<\/em> that does not contain <em>x<\/em>, then <em>F<\/em> contains <em>U<\/em> \u2229<em> V<\/em>, which contains neither point, contradiction.\u00a0 It follows that <em>F<\/em> is exactly the set of open neighborhoods of a unique point.<\/p>\n<p>That finishes to show that <strong>Zh<\/strong>(<em>X<\/em>) is a bounded complete, algebraic poset model of <em>X<\/em>, if <em>X<\/em> is T<sub>1<\/sub>.<\/p>\n<h2>Every T<sub>1<\/sub> space has a dcpo model<\/h2>\n<p>This is due to Zhao and Xi [5,6], and proceeds by embedding <strong>Zh<\/strong>(<em>X<\/em>) into a dcpo in such a way that no new maximal element is added (and so that the topology on those maximal elements is unchanged, too).<\/p>\n<p>Zhao and Xi provide a general construction to achieve that.\u00a0 They start by observing that <strong>Zh<\/strong>(<em>X<\/em>) is a bdcpo (as I call them in Proposition 5.1.60 of the <a href=\"https:\/\/www.cambridge.org\/gb\/knowledge\/isbn\/item7069109\/Non-Hausdorff%20Topology%20and%20Domain%20Theory\/?site_locale=en_GB\">book<\/a>; they call that a local dcpo), namely a poset where every directed family that is bounded from above has a supremum.\u00a0 Then, they show that for every algebraic bdcpo <em>Y<\/em>, one can build a new dcpo <em>Z<\/em> such that Max <em>Y<\/em> and Max <em>Z<\/em> are homeomorphic.\u00a0 Note that <em>Z<\/em> is a dcpo, not just a bdcpo, but we lose algebraicity in the process.<\/p>\n<p>The construction is as follows: <em>Z<\/em> is the set of pairs (<em>y<\/em>, <em>e<\/em>) of elements of <em>Y<\/em> where <em>y<\/em>\u2264<em>e <\/em>and<em> e<\/em> is maximal in<em> Y<\/em>, and those pairs are ordered by (<em>y<\/em>, <em>e<\/em>) \u2264 (<em>y&#8217;<\/em>, <em>e&#8217;<\/em>) if and only if <em>y<\/em>\u2264<em>y&#8217;,<\/em> and either <em>e=e&#8217;<\/em> or <em>y&#8217;=e&#8217;<\/em>.\u00a0 That ordering is of course very strange, and the proof that <em>Z<\/em> is indeed a dcpo and that Max <em>Y<\/em> and Max <em>Z<\/em> are homeomorphic is a tedious verification.<\/p>\n<p>If <em>Y<\/em>=<strong>Zh<\/strong>(<em>X<\/em>), then <em>Z<\/em> consists of pairs (<em>F<\/em>, <em>x<\/em>) of a filter <em>F<\/em> of open sets and of a point that belongs to every element of <em>F<\/em>.\u00a0 That is very natural.\u00a0 But the ordering is somehow weird, and reminiscent of Johnstone&#8217;s counterexample (Exercise 5.2.15 in the <a href=\"https:\/\/www.cambridge.org\/gb\/knowledge\/isbn\/item7069109\/Non-Hausdorff%20Topology%20and%20Domain%20Theory\/?site_locale=en_GB\">book<\/a>).\u00a0 If you think of elements of <em>Z<\/em> as being organized in columns, where (<em>F<\/em>, <em>x<\/em>) is in column <em>x<\/em>, then each column <em>x<\/em> has an element &#8220;at infinity&#8221;, which you may picture as lying right at the top of the column: this is (\u03b7(<em>x<\/em>)<em>,<\/em> <em>x<\/em>), writing \u03b7(<em>x<\/em>) for the filter of open neighborhoods of <em>x<\/em>.\u00a0 (This is really a maximal element if <em>X<\/em> is T<sub>1.<\/sub>)\u00a0 To go up from (<em>F<\/em>, <em>x<\/em>), you may either go up inside the current column, or you may move directly to the point at infinity of a different column, say <em>x&#8217;<\/em>, where <em>x&#8217;<\/em> is another point that belongs to every element of <em>F<\/em>.<\/p>\n<p>The resulting dcpo <em>Z<\/em> provides a dcpo model for <em>X<\/em>, for whichever space <em>X<\/em>, as long as it is T<sub>1<\/sub>.\u00a0 We obtain a dcpo, but we lose algebraicity, and bounded completeness.<\/p>\n<h2>Not every T<sub>1<\/sub> space has a model that is both a dcpo and bounded complete<\/h2>\n<p>Those constructions tend to suggest that there should be a more elegant construction of a model of <em>X<\/em> that would be both a dcpo and bounded complete.<\/p>\n<p>That is hopeless, as Zhao and Xi showed [1]: the set <strong>N<\/strong> of natural numbers, with the cofinite topology, does not have <em>any<\/em> bounded complete dcpo model.<\/p>\n<p>At some point, I will need the following result, which I am stating and proving right now, so that we won&#8217;t have to take a detour later.<\/p>\n<p><strong>Fact.<\/strong> In a topology space, every directed family is irreducible, in the sense that if it is included in a finite union of closed sets, then it is included in one of them.<\/p>\n<p>Proof. Let <em>D<\/em> be a directed family.\u00a0 It is non-empty.\u00a0 Let\u00a0<em>C<\/em>, <em>C&#8217;<\/em> be two closed sets.\u00a0 If <em>D<\/em> is included in <em>C<\/em> \u222a <em>C&#8217;<\/em>, but not in <em>C<\/em> and not in <em>C&#8217;<\/em>, then there is a point <em>x<\/em> of <em>D<\/em> that is not in <em>C<\/em>, a point <em>x&#8217;<\/em> of <em>D<\/em> that is not in <em>C&#8217;<\/em>.\u00a0 By directedness, there is a point <em>y<\/em> above both <em>x<\/em> and <em>x&#8217;<\/em>.\u00a0 Since closed sets are downwards-closed, <em>y<\/em> is neither in <em>C<\/em> nor in <em>C&#8217;<\/em>, hence not in <em>C<\/em> \u222a <em>C&#8217;<\/em>, contradiction.\u00a0 \u2610<\/p>\n<p>Zhao and Xi start by showing some properties of spaces of maximal elements of bounded complete dcpos.\u00a0 Let <em>Y<\/em> be a bounded complete dcpo. For every pair of elements <em>y<\/em>, <em>y&#8217;<\/em> of <em>Y<\/em>, say that <em>y<\/em> and <em>y&#8217;<\/em> are <em>friends<\/em> if and only if they have a common upper bound.\u00a0 (A more usual name would be &#8220;compatible&#8221; or &#8220;coherent&#8221;.\u00a0 The name &#8220;friends&#8221; is mine, and for the present purpose. Zhao and Xi also use an equivalent definition, but which I find harder to grasp.)<\/p>\n<p><strong>Lemma.<\/strong>\u00a0 Let <em>Y<\/em> be a bounded complete dcpo, and\u00a0<em>y<\/em> be a point of\u00a0<em>Y<\/em>.\u00a0 The set of friends of\u00a0<em>y<\/em> is Scott-closed.<\/p>\n<p><em>Proof.<\/em>\u00a0 It is clearly downwards-closed.\u00a0 Let (<em>y<sub>i<\/sub><\/em>)<em><sub>i \u2208 I<\/sub><\/em> be a directed family of friends of <em>y<\/em>.\u00a0 Since <em>Y<\/em> is a bc-domain, for each <em>i<\/em>, <em>y<\/em> and\u00a0<em>y<sub>i<\/sub><\/em> not only have an upper bound, but a least upper bound <em>y<\/em>\u22c1<em>y<sub>i<\/sub><\/em>.\u00a0 The family\u00a0(<em>y<\/em>\u22c1<em>y<sub>i<\/sub><\/em>)<em><sub>i \u2208 I<\/sub><\/em> is directed, and its supremum is an upper bound of both <em>y<\/em> and of sup<em><sub>i \u2208 I<\/sub><\/em> <em>y<sub>i<\/sub><\/em>.\u00a0\u00a0 Hence <em>y<\/em> and sup<em><sub>i \u2208 I<\/sub><\/em> <em>y<sub>i<\/sub><\/em> are friends. \u00a0\u2610<\/p>\n<p><strong>Corollary.<\/strong> Let <em>Y<\/em> be a bounded complete dcpo, <em>X<\/em>=Max <em>Y<\/em>, and\u00a0<em>y<\/em> be a point of\u00a0<em>Y<\/em>.\u00a0 Then <em>X<\/em> \u2229 \u2191<em>y<\/em> is closed in <em>X<\/em>.<\/p>\n<p>Indeed, <em>X<\/em> \u2229 \u2191<em>y<\/em> is just the set of maximal elements of <em>Y<\/em> that are friends with <em>y<\/em>.<\/p>\n<p>Now we are ready for Zhao and Xi&#8217;s counterexample.<\/p>\n<p>Let <strong>N<\/strong> be the set of natural numbers, with the cofinite topology.\u00a0 (Any infinite set would fit equally well.)\u00a0 Recall that its closed sets are <strong>N<\/strong> itself, plus all the finite subsets of <strong>N<\/strong>.<\/p>\n<p>Assume <strong>N<\/strong> had a bounded complete dcpo model<em> Y<\/em>.\u00a0 Equate <strong>N<\/strong> with Max <em>Y<\/em>.\u00a0 Let <em>A<\/em> be the complement of {0} in <strong>N<\/strong>.\u00a0 <em>A<\/em> is not closed in <strong>N<\/strong>.\u00a0 Let <em>B<\/em>=\u2193<em>A<\/em> be the downward closure of <em>A<\/em> in <em>Y<\/em>.\u00a0 Since <em>A<\/em>=<strong>N<\/strong> \u2229 <em>B<\/em>, <em>B<\/em> cannot be (Scott-)closed, since otherwise <em>A<\/em> would be closed in Max <em>Y<\/em>=<strong>N<\/strong>.\u00a0 Therefore, there is a directed family (<em>y<sub>i<\/sub><\/em>)<em><sub>i \u2208 I<\/sub><\/em> in <em>B<\/em> whose supremum <em>y<\/em> is not in <em>B<\/em>.\u00a0 By definition, <em>y<\/em> is not below any element of <strong>N<\/strong> except 0.\u00a0 In other words, <em>y<\/em> is below 0 and, say, not below 1.<\/p>\n<p>Since <em>y<\/em> is not below 1, some <em>y<sub>i<\/sub><\/em> is not below 1 either.\u00a0 (The set of elements below 1, or in general below any fixed element, is Scott-closed.)\u00a0 Look at the set <em>C<\/em>=<strong>N<\/strong> \u2229 \u2191<em>y<sub>i<\/sub><\/em> of friends of <em>y<sub>i<\/sub><\/em> that are in <strong>N<\/strong>.\u00a0 By the corollary stated above, <em>C<\/em> is closed in <strong>N<\/strong>.\u00a0 It cannot be the whole of <strong>N<\/strong>, because <em>y<sub>i<\/sub><\/em> is not below 1, which means that 1 is not in <em>C<\/em>.\u00a0 Hence <em>C<\/em> must be finite.\u00a0 Poor guy, that <em>y<sub>i<\/sub><\/em>: it only has finitely many natural number friends.<\/p>\n<p>It might even be that <em>y<sub>i<\/sub><\/em> has no friend at all&#8230; or can it?\u00a0 No, fortunately, <em>y<sub>i<\/sub><\/em> can count on 0.\u00a0 Indeed, <em>y<sub>i<\/sub><\/em> is below <em>y<\/em>, which is below 0, so\u00a0<em>y<sub>i<\/sub><\/em> and 0 <em>are<\/em> friends.<\/p>\n<p>Now look at the set <em>A&#8217;<\/em> of its other natural number friends: the finite set of natural numbers above <em>y<sub>i<\/sub><\/em>, but different from 0.\u00a0 For every <em>j<\/em> in <em>I<\/em>, by directedness there is an\u00a0<em>y<sub>k<\/sub><\/em> above both <em>y<sub>j<\/sub><\/em> and <em>y<sub>i<\/sub><\/em>; above <em>y<sub>k<\/sub><\/em> one finds an element <em>a<\/em> of <em>A<\/em>, since every <em>y<sub>k<\/sub><\/em> was taken from <em>B<\/em>=\u2193<em>A<\/em>.\u00a0 By definition of <em>A<\/em>, <em>a<\/em>\u22600, and since <em>a<\/em> is also above <em>y<sub>i<\/sub><\/em>, <em>a<\/em> is in <em>A&#8217;<\/em>.\u00a0 It follows that every\u00a0<em>y<sub>j<\/sub><\/em> is in \u2193<em>A&#8217;<\/em>.<\/p>\n<p>Remember the fact we proved as a preliminary step: every directed family is irreducible.\u00a0 Writing \u2193<em>A&#8217;<\/em> as a finite union of closed sets \u2193<em>a<\/em>, <em>a<\/em> in <em>A&#8217;<\/em>, we obtain that the family (<em>y<sub>i<\/sub><\/em>)<em><sub>i \u2208 I<\/sub><\/em> lies entirely inside \u2193<em>a<\/em>, for some <em>a<\/em> in <em>A&#8217;<\/em>.\u00a0 Then its supremum <em>y<\/em> must also be in \u2193<em>a<\/em>.\u00a0 However, <em>a<\/em>\u22600&#8230; but <em>y<\/em> is not below any element of <strong>N<\/strong> except 0: contradiction.<\/p>\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>(April 28th, 2017)<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<ol>\n<li>On topological spaces that have a bounded complete DCPO model.\u00a0\u00a0<a href=\"https:\/\/projecteuclid.org\/euclid.rmjm\">Rocky Mountain Journal of Mathematics<\/a>, Volume 48, Number 1 (2018), 141-156. \u00a0 Note: the copy that D. Zhao gave me is entitled &#8216;On topological spaces that have a bounded complete dcpo model&#8217;.\u00a0 (Thanks to him!)\u00a0 It also contains additional results that I have not mentioned here.<\/li>\n<li>Carl Mummert and Frank Stephan.\u00a0 Topological aspects of poset spaces.\u00a0 Michigan Mathematical Journal, 59, 2010, pages 3-24.<\/li>\n<li>Dongsheng Zhao.\u00a0 Poset models of topological spaces.\u00a0 In Proc. Intl. Conf. Quantitative Logic and Quantification of Software, Global Link publisher, 2009, pages 229-238.<\/li>\n<li>Marcel Ern\u00e9.\u00a0 Algebric models for T<sub>1<\/sub> spaces.\u00a0 Topology and its Applications 158(7), 2011, pages 945-962.<\/li>\n<li>Xiaoyong Xi and Dongsheng Zhao.\u00a0 Well-filtered spaces and their dcpo models.\u00a0 Mathematical Structures in Computer Science 27(4), may 2017, pages 507-515. DOI:<a href=\"https:\/\/dx.doi.org\/10.1017\/S0960129515000171\"> https:\/\/dx.doi.org\/10.1017\/S0960129515000171<\/a><\/li>\n<li>Dongsheng Zhao and Xiaoyong Xi.\u00a0 <a href=\"https:\/\/www.cambridge.org\/core\/journals\/mathematical-proceedings-of-the-cambridge-philosophical-society\/article\/directed-complete-poset-models-of-t-1-spaces\/298A48A900F76FDD02E878A698906A31\">Directed complete poset models of T<sub>1<\/sub> spaces<\/a>.\u00a0 Mathematical Proceedings of the Cambridge Philosophical Society, october 2016. DOI: <a class=\"url\" href=\"https:\/\/doi.org\/10.1017\/S0305004116000888\" target=\"_blank\" rel=\"noopener noreferrer\">https:\/\/doi.org\/10.1017\/S0305004116000888<\/a><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Dongsheng Zhao recently mentioned one recent result of his and Xiaoyong Xi to me, on models of spaces [1], which leads to a curious situation. Let us call model of a topological space X any poset Y whose subset of &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1184\">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-1184","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1184","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=1184"}],"version-history":[{"count":17,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1184\/revisions"}],"predecessor-version":[{"id":5939,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1184\/revisions\/5939"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1184"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}