{"id":838,"date":"2016-03-10T09:14:53","date_gmt":"2016-03-10T08:14:53","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=838"},"modified":"2022-05-17T09:36:52","modified_gmt":"2022-05-17T07:36:52","slug":"ideal-domains-iii-quasi-ideal-models","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=838","title":{"rendered":"Ideal domains III: Quasi-ideal models"},"content":{"rendered":"<p>I am a bit stubborn. In my first post on <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=681\">ideal domains<\/a>, I thought I would be able to extend Keye Martin\u2019s result from metric to quasi-metric spaces. I have said I had failed, but now I think I have succeeded.\u00a0 This leads to a notion that I will call a <em>quasi-ideal domain<\/em>.<\/p>\n<p>Our purpose today is to show that, if <em>X<\/em> is a continuous Yoneda-complete quasi-metric space, then it embeds into an algebraic dcpo, and in fact, in a very specific way: as the subspace of limit elements of a quasi-ideal domain.<\/p>\n<p>The elements of that quasi-ideal domain will again be certain formal balls, and will look very much like Keye Martin&#8217;s <strong>B<\/strong><em><sub>D<\/sub><\/em>(<em>X<\/em>, <em>d<\/em>). If you refer to the <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=681\">first post<\/a>, the main difference will be that our ordering \u2291 will be such that (<em>x<\/em>, 0) \u2291 (<em>y<\/em>, 0) whenever <em>x<\/em> \u2264 <em>y<\/em> \u2014 that was not the case before.<\/p>\n<h2>Embedding continuous Yoneda-complete spaces into algebraic domains<\/h2>\n<p>Let us therefore define (<em>x<\/em>, <em>r<\/em>) \u228f (<em>y<\/em>, <em>s<\/em>) iff:<\/p>\n<ul>\n<li>either (<em>x<\/em>, <em>r<\/em>) \u226a (<em>y<\/em>, <em>s<\/em>) and <em>r \u2265 2s<\/em><\/li>\n<li>or <em>r<\/em>=<em>s<\/em>=0 and <em>x<\/em> \u2264<em> y<\/em> (i.e.,<em>d<\/em>(<em>x<\/em>, <em>y<\/em>)<em>=<\/em>0, or equivalently, (<em>x<\/em>, 0) \u2264 (<em>y<\/em>, 0)).<em><br \/>\n<\/em><\/li>\n<\/ul>\n<p>Note that I wrote \u228f, not \u2291: \u228f is a kind of strict part of \u2291. Now define (<em>x<\/em>, <em>r<\/em>) \u2291 (<em>y<\/em>, <em>s<\/em>) iff (<em>x<\/em>, <em>r<\/em>) \u228f (<em>y<\/em>, <em>s<\/em>) or (<em>x<\/em>, <em>r<\/em>) = (<em>y<\/em>, <em>s<\/em>). With that correction, \u2291 is reflexive, and a painful case analysis shows that it is transitive. Also, \u2291 implies \u2264, so the relation is antisymmetric, hence an ordering. Fine.<\/p>\n<p>Let me reuse our former name, and call <strong>B<\/strong><em><sub>D<\/sub><\/em>(<em>X<\/em>) the set of formal balls with the ordering \u2291.<\/p>\n<p><strong>Lemma.<\/strong> For every Yoneda-complete quasi-metric space <em>X<\/em>, <strong>B<\/strong><em><sub>D<\/sub><\/em>(<em>X<\/em>) is a dcpo, and in fact one in which directed suprema are computed exactly as in <strong>B<\/strong>(<em>X<\/em>).<\/p>\n<p><em>Proof.<\/em> There is a similar lemma in <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=681\">Ideal Models I<\/a>, namely the very first one, although it only applied to complete metric spaces. I had mentioned that the only place where we needed <em>X<\/em> to be metric was the last line, and this misfortune is now repaired by our new definition of \u2291.<\/p>\n<p>We take a family (<em>x<sub>i<\/sub><\/em>, <em>r<sub>i<\/sub><\/em>)<em><sub>i \u2208 I<\/sub><\/em> in <strong>B<\/strong><em><sub>D<\/sub><\/em>(<em>X<\/em>) that is directed with respect to \u2291, and we let (<em>x<\/em>, <em>r<\/em>) = (<em>d<\/em>-lim <em>x<sub>i<\/sub><\/em>, inf <em>r<sub>i<\/sub><\/em>) be its supremum in <strong>B<\/strong>(<em>X<\/em>). (We are using the Kostanek-Waszkiewicz Theorem here.) We must show that (<em>x<\/em>, <em>r<\/em>) is also its supremum with respect to \u2291. If the supremum is reached, that is clear. Otherwise, we still have a few cases to distinguish. If <em>r<sub>i<\/sub><\/em>=0 for <em>i<\/em> large enough, then we can consider the subfamily of formal balls (<em>x<sub>i<\/sub><\/em>, <em>r<sub>i<\/sub><\/em>) with <em>r<sub>i<\/sub><\/em>=0, and since \u2291 and \u2264 coincide on them, we are done.<\/p>\n<p>The final case is where <em>r<sub>i<\/sub><\/em>&gt;0 for every <em>i<\/em>. In that case, using the same trick as <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=681\">Ideal Models I<\/a>, every (<em>x<sub>i<\/sub><\/em>, <em>r<sub>i<\/sub><\/em>) is \u228f some (<em>x<sub>j<\/sub><\/em>, <em>r<sub>j<\/sub><\/em>). This implies that <em>r<\/em>=0, and also that (<em>x<\/em>, <em>r<\/em>)=(<em>x<\/em>, 0) is an upper bound of the family (with respect to \u2291): for every <em>i<\/em>, (<em>x<sub>i<\/sub><\/em>, <em>r<sub>i<\/sub><\/em>) \u228f (<em>x<sub>j<\/sub><\/em>, <em>r<sub>j<\/sub><\/em>) \u2264 (<em>x<\/em>, <em>r<\/em>) = (<em>x<\/em>, 0), so (<em>x<sub>i<\/sub><\/em>, <em>r<sub>i<\/sub><\/em>) \u226a (<em>x<\/em>, 0), namely (<em>x<sub>i<\/sub><\/em>, <em>r<sub>i<\/sub><\/em>) \u228f (<em>x<\/em>, <em>r<\/em>) by the first clause of the definition of \u228f.<\/p>\n<p>If (<em>y<\/em>, <em>s<\/em>) is another upper bound of (<em>x<sub>i<\/sub><\/em>, <em>r<sub>i<\/sub><\/em>)<em><sub>i \u2208 I<\/sub><\/em> (with respect to \u2291), then it is also an upper bound with respect to \u2264, so (<em>x<\/em>, <em>r<\/em>) \u2264 (<em>y<\/em>, <em>s<\/em>). In particular, <em>s<\/em>=0. Since also <em>r<\/em>=0, using the second clause of the definition of \u228f, (<em>x<\/em>, <em>r<\/em>) \u2291 (<em>y<\/em>, <em>s<\/em>). \u2610<\/p>\n<p><strong>Lemma.<\/strong> For every continuous Yoneda-complete quasi-metric space <em>X<\/em>, <em>d<\/em>, the finite elements of <strong>B<\/strong><em><sub>D<\/sub><\/em>(<em>X<\/em>) are those of the form (<em>x<\/em>, <em>r<\/em>), <em>r<\/em>\u22600.<\/p>\n<p>Recall that a quasi-metric space is <em>continuous<\/em> Yoneda-complete if and only if its (ordinary) poset of formal balls is a continuous dcpo.<\/p>\n<p>The proof is the same as for the second and third lemmas of <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=681\">Ideal Models I<\/a>. For later reference, we recall how we show that no element (<em>x<\/em>, 0) is finite in <strong>B<\/strong><em><sub>D<\/sub><\/em>(<em>X<\/em>). Since <em>X<\/em> is continuous Yoneda-complete, (<em>x<\/em>, 0) is the supremum of a directed family <em>D<\/em> = (<em>x<sub>i<\/sub><\/em>, <em>r<sub>i<\/sub><\/em>)<em><sub>i \u2208 I<\/sub><\/em> in <strong>B<\/strong>(<em>X<\/em>) with (<em>x<sub>i<\/sub><\/em>, <em>r<sub>i<\/sub><\/em>) \u226a (<em>x<\/em>, 0) for each <em>i<\/em>. (We shall need to remember that this implies <em>r<sub>i<\/sub><\/em>&gt;0.) The way-below relation \u226a is the one from <strong>B<\/strong>(<em>X<\/em>). The family <em>D<\/em> is directed in <strong>B<\/strong>(<em>X<\/em>) (with respect to \u2264) but not necessarily with respect to \u2291. We build another family with the same supremum, which is directed with respect to \u2291. This is a family of formal balls (<em>x<sub>I<\/sub><\/em>, <em>r<sub>I<\/sub><\/em>), where <em>I<\/em> ranges over the finite subsets of <em>D<\/em>. Each is an element of <em>D<\/em>, and is above every element of <em>I<\/em> in the \u2291 ordering. It is also above (<em>x<sub>J<\/sub><\/em>, <em>r<sub>J<\/sub><\/em>) in the \u2291 ordering, for every proper subset <em>J<\/em> of <em>I<\/em>, and that ensures that the family of formal balls (<em>x<sub>I<\/sub><\/em>, <em>r<sub>I<\/sub><\/em>) is directed with respect to \u2291. Once this is done, if (<em>x<\/em>, 0) were finite in <strong>B<\/strong><em><sub>D<\/sub><\/em>(<em>X<\/em>), we would have (<em>x<\/em>, 0) \u2291 (<em>x<sub>I<\/sub><\/em>, <em>r<sub>I<\/sub><\/em>) for some <em>I<\/em>, and that is impossible since <em>r<sub>I<\/sub><\/em>&gt;0.<\/p>\n<p><strong>Theorem.<\/strong> Let <em>X<\/em>, <em>d<\/em> be a continuous Yoneda-complete quasi-metric space. Then <em>X<\/em>, with the <em>d<\/em>-Scott topology, is homeomorphic to the subspace of non-finite elements of the algebraic dcpo <strong>B<\/strong><em><sub>D<\/sub><\/em>(<em>X<\/em>).<\/p>\n<p><em>Proof.<\/em> The previous lemma shows that the non-finite elements of <strong>B<\/strong><em><sub>D<\/sub><\/em>(<em>X<\/em>) are the formal balls (<em>x<\/em>, 0), which are in bijection with the points <em>x<\/em> of <em>X<\/em>. By the paragraph right before the statement of the theorem, (<em>x<\/em>, 0) is the supremum of a family of formal balls, with non-zero radii, and which is directed in <strong>B<\/strong><em><sub>D<\/sub><\/em>(<em>X<\/em>), namely with respect to \u2291. That shows that <strong>B<\/strong><em><sub>D<\/sub><\/em>(<em>X<\/em>) is algebraic.<\/p>\n<p>The rest of the proof is exactly the same as in the corresponding theorem in <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=681\">Ideal Models I<\/a>. Let us call, temporarily, the <em>D<\/em>-Scott topology the topology induced by the inclusion of <em>X<\/em> into <strong>B<\/strong><em><sub>D<\/sub><\/em>(<em>X<\/em>).<\/p>\n<p>An open subset <em>U<\/em> of <em>X<\/em> in the <em>d<\/em>-Scott topology is the intersection of <em>X<\/em> with a Scott-open subset <em>V<\/em> of <strong>B<\/strong>(<em>X<\/em>). Since \u2291 implies \u2264, and since directed suprema are computed in <strong>B<\/strong><em><sub>D<\/sub><\/em>(<em>X<\/em>) as in <strong>B<\/strong>(<em>X<\/em>), <em>V<\/em> is Scott-open in <strong>B<\/strong><em><sub>D<\/sub><\/em>(<em>X<\/em>), hence <em>U<\/em> is <em>D<\/em>-Scott open.<\/p>\n<p>Conversely, let <em>U<\/em> be a <em>D<\/em>-Scott open subset of <em>X<\/em>. For every <em>x<\/em> \u2208 <em>U<\/em>, there is a formal ball (<em>y<\/em>, <em>r<\/em>) \u2291 (<em>x<\/em>, 0) with <em>r<\/em>\u22600 such that every formal ball (<em>z<\/em>, <em>0<\/em>) such that (<em>y<\/em>, <em>r<\/em>) \u2291 (<em>z<\/em>, 0) is in <em>U<\/em>. Since <em>r<\/em>\u22600, (<em>y<\/em>, <em>r<\/em>) \u226a (<em>x<\/em>, 0). For every element <em>z<\/em> such that (<em>y<\/em>, <em>r<\/em>) \u226a (<em>z<\/em>, 0), we plainly observe that (<em>y<\/em>, <em>r<\/em>) \u2291 (<em>z<\/em>, 0), so (<em>z<\/em>, 0) is in <em>U<\/em>. It follows that the intersection of <em>X<\/em> with \u219f(<em>y<\/em>, <em>r<\/em>) contains (<em>x<\/em>, 0) and is included in <em>U<\/em>, showing that <em>U<\/em> is <em>d<\/em>-Scott-open. \u2610<\/p>\n<h2>Quasi-ideal domains<\/h2>\n<p>As we shall see, <strong>B<\/strong><em><sub>D<\/sub><\/em>(<em>X<\/em>) has extra properties that make it look even more like Keye Martin&#8217;s ideal domains.<\/p>\n<p>Let me make a parenthesis, which might otherwise act as a motivation for the whole construction.<\/p>\n<p><a title=\"Daniele Varacca\" href=\"https:\/\/www.lacl.fr\/~dvaracca\/\">Daniele Varacca<\/a> once asked me whether the following notion had a name: algebraic domains where every element below a finite element is itself finite. That seems like it should be a natural notion, but I don&#8217;t think I&#8217;ve seen that anywhere in the literature. The closest is the notion of a dI-domain, where we require a lot more: in a dI-domain, every finite point must have only finitely many points below it\u2014in particular those points will all be finite; a dI-domain is also required to be bounded-complete algebraic, with the property that binary suprema of bounded pairs distribute over binary meets, and we will certainly not make any of those assumptions.<\/p>\n<p>Let me call <strong>quasi-ideal domain<\/strong> any algebraic dcpo in which every element below a finite element is finite. Clearly, every ideal domain satisfies that property. In a quasi-ideal domain, the non-finite elements may fail to be maximal, as the following example shows.<\/p>\n<p><strong>Fact.<\/strong> For every set <em>A<\/em>, the powerset <strong>P<\/strong>(<em>A<\/em>) is quasi-ideal domain. If <em>A<\/em> is infinite, then <strong>P<\/strong>(<em>A<\/em>) is not an ideal domain.<\/p>\n<p>(The finite elements are the finite subsets of <em>A<\/em>, and certainly any subset of <em>A<\/em> that is included in a finite subset is itself finite.)<\/p>\n<p>Anyway, a quasi-ideal domain has just <em>two<\/em> layers: a lower layer of finite elements, and an upper layer of non-finite elements which can all be obtained as directed suprema of elements from the lower layer.<\/p>\n<p>I will call the non-finite elements the <em>limit elements<\/em> of the quasi-ideal domain.<\/p>\n<p>As you can see, <strong>B<\/strong><em><sub>D<\/sub><\/em>(<em>X<\/em>) is one example of such a quasi-ideal domain: if (<em>x<\/em>, <em>r<\/em>) \u2291 (<em>y<\/em>, <em>s<\/em>) where (<em>y<\/em>, <em>s<\/em>) is finite (i.e., <em>s<\/em>\u22600), then (<em>x<\/em>, <em>r<\/em>) \u2264 (<em>y<\/em>, <em>s<\/em>), whence <em>r<\/em> \u2265 <em>s<\/em> &gt; 0, and therefore (<em>x<\/em>, <em>r<\/em>) is finite as well.<\/p>\n<p>We can then rephrase the previous theorem under the following much stronger form.<\/p>\n<p><strong>Theorem.<\/strong> Let <em>X<\/em>, <em>d<\/em> be a continuous Yoneda-complete quasi-metric space. Then <em>X<\/em>, with the <em>d<\/em>-Scott topology, is homeomorphic to the subspace of limit elements of the quasi-ideal domain <strong>B<\/strong><em><sub>D<\/sub><\/em>(<em>X<\/em>).<\/p>\n<h2>\u03c9-continuous Yoneda-complete quasi-metric spaces<\/h2>\n<p>Can we refine all that when <em>X<\/em> is second-countable in its <em>d<\/em>-Scott topology? I do not know yet, but we can if we assume a stronger property, which (by analogy with our notion of continuity for Yoneda-complete quasi-metric spaces) I will call <em>\u03c9-continuity<\/em>.<\/p>\n<p>Call a Yoneda-complete quasi-metric space <em>X<\/em>, <em>d<\/em> <em>\u03c9-continuous<\/em> if and only if its dcpo of formal balls is&#8230; \u03c9-continuous.<\/p>\n<p>Recall that a dcpo is called \u03c9-continuous if and only if it has a countable basis. By Norberg&#8217;s Lemma 7.7.13, this is equivalent to its Scott topology being second-countable. In particular, every \u03c9-continuous Yoneda-complete quasi-metric space is both continuous Yoneda-complete and second-countable in its <em>d<\/em>-Scott topology.\u00a0 (I have not been able to show that, conversely, every second-countable continuous Yoneda-complete quasi-metric space is \u03c9-continuous, and I would be surprised if that were true.)<\/p>\n<p>Now we can refine everything we have done earlier. Fix a basis <em>B<\/em> of <strong>B<\/strong>(<em>X<\/em>, <em>d<\/em>), which typically will be countable for an \u03c9-continuous Yoneda-complete quasi-metric space <em>X<\/em>, <em>d<\/em>. Define <strong>B<\/strong>&#8216;<em><sub>D<\/sub><\/em>(<em>X<\/em>, <em>d<\/em>) as the following variant of <strong>B<\/strong><em><sub>D<\/sub><\/em>(<em>X<\/em>, <em>d<\/em>):<\/p>\n<ul>\n<li>The elements of <strong>B<\/strong>&#8216;<em><sub>D<\/sub><\/em>(<em>X<\/em>, <em>d<\/em>) are the formal balls in the basis <em>B<\/em>, plus the formal balls of the form (<em>x<\/em>, 0), <em>x<\/em> \u2208 <em>X<\/em>.<\/li>\n<li>The ordering \u2291 is defined as before: (<em>x<\/em>, <em>r<\/em>) \u2291 (<em>y<\/em>, <em>s<\/em>) iff (<em>x<\/em>, <em>r<\/em>) \u228f (<em>y<\/em>, <em>s<\/em>) or (<em>x<\/em>, <em>r<\/em>) = (<em>y<\/em>, <em>s<\/em>), where (<em>x<\/em>, <em>r<\/em>) \u228f (<em>y<\/em>, <em>s<\/em>) iff either (<em>x<\/em>, <em>r<\/em>) \u226a (<em>y<\/em>, <em>s<\/em>) and <em>r \u2265 2s<\/em>, or <em>r<\/em>=<em>s<\/em>=0 and <em>x<\/em> \u2264<em> y<\/em>.<\/li>\n<\/ul>\n<p><strong>Lemma.<\/strong> For every \u03c9-continuous Yoneda-complete quasi-metric space <em>X<\/em>, <strong>B<\/strong>&#8216;<em><sub>D<\/sub><\/em>(<em>X<\/em>) is a dcpo, and in fact one in which directed suprema are computed exactly as in <strong>B<\/strong>(<em>X<\/em>).<\/p>\n<p>This is proved exactly as our first lemma on <strong>B<\/strong><em><sub>D<\/sub><\/em>(<em>X<\/em>). The important new thing to check is that, given a family (<em>x<sub>i<\/sub><\/em>, <em>r<sub>i<\/sub><\/em>)<em><sub>i \u2208 I<\/sub><\/em> in <strong>B<\/strong>&#8216;<em><sub>D<\/sub><\/em>(<em>X<\/em>) that is directed with respect to \u2291, then its sup in <strong>B<\/strong>(<em>X<\/em>) will always be in <strong>B<\/strong>&#8216;<em><sub>D<\/sub><\/em>(<em>X<\/em>), that is, it will either be in the basis <em>B<\/em> or have radius 0. This follows from the case analysis we have already done in our previous similar lemma.<\/p>\n<p><strong>Lemma.<\/strong> For every \u03c9-continuous Yoneda-complete quasi-metric space <em>X<\/em>, <em>d<\/em>, the finite elements of <strong>B<\/strong>&#8216;<em><sub>D<\/sub><\/em>(<em>X<\/em>) are those of the form (<em>x<\/em>, <em>r<\/em>), <em>r<\/em>\u22600, namely those of the basis <em>B<\/em>.<\/p>\n<p>That is again proved as before.<\/p>\n<p>As a corollary, we obtain the following variant of our first Theorem: Let <em>X<\/em>, <em>d<\/em> be a continuous Yoneda-complete quasi-metric space, and fix a basis <em>B<\/em> of its dcpo of formal balls. Then <em>X<\/em>, with the <em>d<\/em>-Scott topology, is homeomorphic to the subspace of non-finite elements of the algebraic dcpo <strong>B<\/strong><em>&#8216;<\/em><em><sub>D<\/sub><\/em>(<em>X<\/em>), and the finite elements of the latter are exactly the formal balls in <em>B<\/em> with non-zero radius. That is, again, proved as before.<\/p>\n<p>When <em>B<\/em> is countable, that implies that <strong>B<\/strong>&#8216;<em><sub>D<\/sub><\/em>(<em>X<\/em>) has countably many finite elements:<\/p>\n<p><strong>Theorem.<\/strong> Let <em>X<\/em>, <em>d<\/em> be an \u03c9-continuous Yoneda-complete quasi-metric space. Then <em>X<\/em>, with the <em>d<\/em>-Scott topology, is homeomorphic to the subspace of non-finite elements of the \u03c9-algebraic dcpo <strong>B<\/strong><em><sub>D<\/sub><\/em>(<em>X<\/em>).<\/p>\n<p>Let us rephrase that in the language of ideal completion remainders (see <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=808\">Ideal Models II<\/a>): Every \u03c9-continuous Yoneda-complete quasi-metric space is the ideal completion remainder of a countable poset.<\/p>\n<p>We have seen that the ideal completion remainders of countable posets are the quasi-Polish spaces (a theorem due to Matthew de Brecht). We therefore obtain:<\/p>\n<p><strong>Theorem.<\/strong> The topological spaces underlying the \u03c9-continuous Yoneda-complete quasi-metric spaces (with the <em>d<\/em>-Scott topology) are exactly the same as those underlying the separable Smyth-complete quasi-metric spaces (with the open ball topology \u2014 which in fact coincides with the <em>d<\/em>-Scott topology in that case), namely the quasi-Polish spaces.<\/p>\n<p>That finally answers one of the questions I had about the definition of quasi-Polish spaces: why should we consider <em>Smyth-complete<\/em> spaces instead of, say, Yoneda-complete spaces to define them?<\/p>\n<p>The answer is, provided we take \u03c9-continuous Yoneda-complete spaces, and take care of using the <em>d<\/em>-Scott topology (the right topology, in all situations that I know of):<br \/>\nthis <em>does not matter<\/em>.<\/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>\u00a0(March 10th, 2016)<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","protected":false},"excerpt":{"rendered":"<p>I am a bit stubborn. In my first post on ideal domains, I thought I would be able to extend Keye Martin\u2019s result from metric to quasi-metric spaces. I have said I had failed, but now I think I have &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=838\">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":"open","ping_status":"open","template":"","meta":{"_crdt_document":"","footnotes":""},"class_list":["post-838","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/838","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=838"}],"version-history":[{"count":29,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/838\/revisions"}],"predecessor-version":[{"id":5348,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/838\/revisions\/5348"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=838"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}