{"id":2416,"date":"2020-05-20T10:32:26","date_gmt":"2020-05-20T08:32:26","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=2416"},"modified":"2022-11-19T15:03:04","modified_gmt":"2022-11-19T14:03:04","slug":"quasi-polish-spaces-as-rounded-ideal-completions","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=2416","title":{"rendered":"Quasi-Polish spaces as rounded ideal completions"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">The rounded ideal completion <strong>RI<\/strong>(<em>B<\/em>,\u227a) of a transitive, interpolative relation \u227a on a set <em>B<\/em> (an <em>abstract basis<\/em>), is always a continuous dcpo, and every continuous dcpo can be written this way, up to isomorphism.  Quite some time ago, Matthew de Brecht mentioned the following pearl to me: if you remove the interpolation requirement, and ask <i>B<\/i> to be countable, then the rounded ideal completions of transitive relations on a countable set <em>B<\/em> (with a suitable topology) are exactly the quasi-Polish spaces.  This can be found in [1], amidst some computability considerations which I will ignore here.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">The rounded ideal completion<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Let \u227a be a transitive relation on a set <em>B<\/em>.  While the rounded ideals of <em>B<\/em>,\u227a are usually defined when \u227a is not just transitive but also interpolative, the definition makes perfect sense without interpolation.  Let me recall that a <em>rounded ideal<\/em> in <em>B<\/em>,\u227a is a subset <em>I<\/em> of <em>B<\/em> that is:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><em>downwards-closed<\/em>: for all <em>x<\/em>, <em>y<\/em> \u2208 <em>B<\/em> such that <em>x<\/em>\u227a<em>y<\/em> and <em>y<\/em> \u2208 <em>I<\/em>, <em>x<\/em> is in <em>I<\/em>;<\/li>\n\n\n\n<li>and <em>directed<\/em>: for every finite collection of elements <em>x<\/em><sub>1<\/sub>, &#8230;, <em>x<sub>n<\/sub><\/em> of <em>I<\/em>, there is an element <em>x<\/em> of <em>I<\/em> such that <em>x<\/em><sub>1<\/sub>, &#8230;, <em>x<sub>n<\/sub><\/em> \u227a <em>x<\/em>.  When <em>n<\/em>=0, this means that <em>I<\/em> is non-empty.  When <em>n<\/em>=2, this is the usual directedness condition that any two elements of <em>I<\/em> have a common upper bound in <em>I<\/em>.  When <em>n<\/em>=1, this says that for every element <em>x<\/em> of <em>I<\/em>, there is an element <em>y<\/em> of <em>I<\/em> such that <em>x<\/em>\u227a<em>y<\/em>: that would come for free if \u227a were reflexive, but, precisely, \u227a is not reflexive in general.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">It is traditional to equip <strong>RI<\/strong>(<em>B<\/em>,\u227a), the set of rounded ideals of <em>B<\/em>,\u227a, with the Scott topology of the inclusion ordering.  That works marvelously well when <em>B<\/em>,\u227a is an abstract basis, that is, when \u227a is not only transitive but <em>interpolative<\/em>: \u227a is interpolative if and only if for all elements <em>x<\/em><sub>1<\/sub>, &#8230;, <em>x<sub>n<\/sub><\/em> and <em>x<\/em> such that <em>x<\/em><sub>1<\/sub>, &#8230;, <em>x<sub>n<\/sub><\/em> \u227a <em>x<\/em>, there is an element <em>y<\/em> of <em>B<\/em> such that <em>x<\/em><sub>1<\/sub>, &#8230;, <em>x<sub>n<\/sub><\/em> \u227a <em>y<\/em> \u227a <em>x<\/em>.  In that case, <strong>RI<\/strong>(<em>B<\/em>,\u227a) is a continuous dcpo, and conversely, every continuous dcpo <em>X<\/em> can be obtained this way up to isomorphism (Proposition 5.1.33 in the <a href=\"https:\/\/www.cambridge.org\/gb\/knowledge\/isbn\/item7069109\/Non-Hausdorff%20Topology%20and%20Domain%20Theory\/?site_locale=en_GB\"><a href=\"https:\/\/www.cambridge.org\/gb\/knowledge\/isbn\/item7069109\/Non-Hausdorff%20Topology%20and%20Domain%20Theory\/?site_locale=en_GB\">book<\/a><\/a>).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">However, in the case where \u227a is transitive but not necessarily interpolative, we must equip <strong>RI<\/strong>(<em>B<\/em>,\u227a) with another topology [1, Definition 1], which I will call the <em>weak topology<\/em>.  This is the topology generated by the subbasic open sets [<em>x<\/em>], <em>x<\/em> \u2208 <em>B<\/em>, where [<em>x<\/em>] is the collection of rounded ideals that contain <em>x<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The weak topology is always coarser (weaker) than the Scott topology of inclusion, and coincides with it if <em>B<\/em>,\u227a is an abstract basis.  Indeed, by Proposition 5.1.33 of the <a href=\"https:\/\/www.cambridge.org\/gb\/knowledge\/isbn\/item7069109\/Non-Hausdorff%20Topology%20and%20Domain%20Theory\/?site_locale=en_GB\">book<\/a>, a basis of <strong>RI<\/strong>(<em>B<\/em>,\u227a) is given by the sets \u21d3<em>x<\/em> = {<em>y<\/em> \u2208 <em>B<\/em> | <em>y<\/em>\u227a<em>x<\/em>}, <em>x<\/em> \u2208 <em>B<\/em>; the sets \u219f\u21d3<em>x<\/em> = {<em>I<\/em> \u2208 <strong>RI<\/strong>(<em>B<\/em>,\u227a) | \u21d3<em>x<\/em> \u226a <em>I<\/em>} form a base of the Scott topology, and one checks easily that those sets \u219f\u21d3<em>x<\/em> are simply the sets [<em>x<\/em>].<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Henceforth, we will always equip <strong>RI<\/strong>(<em>B<\/em>,\u227a) with the weak topology.  This will cover both the interpolative and the non-interpolative cases.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Embedding <strong>RI<\/strong>(<em>B<\/em>,\u227a) inside a continuous (even algebraic) dcpo<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">We wish to show that, when <em>B<\/em> is countable and \u227a is merely transitive, then <strong>RI<\/strong>(<em>B<\/em>,\u227a) is quasi-Polish.  It suffices to embed it as a some <strong>\u03a0<\/strong><sup>0<\/sup><sub>2<\/sub> subspace (see later) of an \u03c9-continuous dcpo, because every <strong>\u03a0<\/strong><sup>0<\/sup><sub>2<\/sub> subspace of an \u03c9-continuous dcpo is quasi-Polish [2, Corollary 45+Corollary 23].  (I will say that again below.)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Working our way towards that result, will embed <strong>RI<\/strong>(<em>B<\/em>,\u227a), where \u227a is transitive but not necessarily interpolative, inside a continuous dcpo.  This is remarkably simple.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let \u2aaf be the reflexive closure of \u227a.  In other words, <em>x<\/em>\u2aaf<em>y<\/em> if and only if <em>x<\/em>\u227a<em>y<\/em> or <em>x<\/em>=<em>y<\/em>.  We will simply embed <strong>RI<\/strong>(<em>B<\/em>,\u227a) inside <strong>RI<\/strong>(<em>B<\/em>,\u2aaf).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The relation \u2aaf is of course transitive, and it is also trivially interpolative: for all elements <em>x<\/em><sub>1<\/sub>, &#8230;, <em>x<sub>n<\/sub><\/em> and <em>x<\/em> such that <em>x<\/em><sub>1<\/sub>, &#8230;, <em>x<sub>n<\/sub><\/em> \u2aaf <em>x<\/em>, there is an element <em>y<\/em> of <em>B<\/em> such that <em>x<\/em><sub>1<\/sub>, &#8230;, <em>x<sub>n<\/sub><\/em> \u2aaf <em>y<\/em> \u2aaf <em>x<\/em>&#8230; namely <em>x<\/em> itself.  As we noticed earlier, every reflexive relation is interpolative.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Then <strong>RI<\/strong>(<em>B<\/em>,\u2aaf) is a continuous dcpo, as we have recalled above, and its weak topology coincides with the Scott topology of inclusion.  Even better: since \u2aaf is a preordering, <strong>RI<\/strong>(<em>B<\/em>,\u2aaf) is simply the ideal completion <strong>I<\/strong>(<em>B<\/em>,\u2aaf), which is an algebraic dcpo, whose finite elements are the sets \u2193<em>x<\/em> = {<em>y<\/em> \u2208 <em>B<\/em> | <em>y<\/em>\u2aaf<em>x<\/em>}, <em>x<\/em> \u2208 <em>B<\/em>.  The (rounded) ideals of <em>B<\/em>,\u2aaf are simply its ideals.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let us keep the notation [<em>x<\/em>] for the subbasic open subsets of <strong>RI<\/strong>(<em>B<\/em>,\u227a), namely the set of rounded ideals <em>I<\/em> of <em>B<\/em>,\u227a that contain <em>x<\/em>, and let us use the new notation [<em>x<\/em>]&#8217; for the corresponding subbasic open subsets of <strong>RI<\/strong>(<em>B<\/em>,\u2aaf), namely the set of <em>ideals<\/em> <em>I&#8217;<\/em> of <em>B<\/em>,\u2aaf (not \u227a) that contain <em>x<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We consider the function <em>i<\/em> : <strong>RI<\/strong>(<em>B<\/em>,\u227a) \u2192 <strong>RI<\/strong>(<em>B<\/em>,\u2aaf) that maps every rounded ideal <em>I<\/em> of <em><em>B<\/em><\/em>,\u227a to itself, seen as an ideal of <em>B<\/em>,\u2aaf.  I will let you check that this is well-defined, and continuous: the inverse image <em>i<\/em><sup>-1<\/sup>([<em>x<\/em>]&#8217;) is [<em>x<\/em>].<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The map <em>i<\/em> is also <em>almost open<\/em>, namely every open subset <em>U<\/em> of <strong>RI<\/strong>(<em>B<\/em>,\u227a) is the inverse image of some open subset <em>V<\/em> of <strong><strong>RI<\/strong><\/strong>(<em><em>B<\/em><\/em>,\u2aaf).  Indeed, <em>U<\/em> is a union of finite intersections of sets of the form [<em>x<\/em>], and it suffices to take the union of finite intersections of the corresponding sets [<em>x<\/em>]&#8217; for <em>V<\/em>.  Let me recall that the continuous, almost open, and injective maps are exactly the topological embeddings (Proposition 4.10.9 in the <a href=\"https:\/\/www.cambridge.org\/gb\/knowledge\/isbn\/item7069109\/Non-Hausdorff%20Topology%20and%20Domain%20Theory\/?site_locale=en_GB\">book<\/a>).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">And precisely, <em>i<\/em> is injective, as should be clear.  This gives us the desired embedding.  We sum up the results we have obtained so far as follows.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Lemma A.<\/strong>  For every set <em>B<\/em>, for every transitive relation \u227a on <em>B<\/em>, the inclusion map <em>i<\/em> : <strong>RI<\/strong>(<em>B<\/em>,\u227a) \u2192 <strong>RI<\/strong>(<em><em>B<\/em><\/em>,\u2aaf) is a topological embedding, and <strong>RI<\/strong>(<em><em>B<\/em><\/em>,\u2aaf) is an algebraic dcpo with the Scott topology of inclusion.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Characterizing the image of <em>i<\/em><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">When <em>B<\/em> is countable, <strong>RI<\/strong>(<em><em>B<\/em><\/em>,\u2aaf) is \u03c9-continuous (even \u03c9-algebraic): that simply means that it has a countable basis, and that basis is just <em>B<\/em>.  We need to show that its subspace <strong>RI<\/strong>(<em>B<\/em>,\u227a) has a special shape, namely that it is a <strong>\u03a0<\/strong><sup>0<\/sup><sub>2<\/sub> subspace (again, see later).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">To this end, we characterize <strong>RI<\/strong>(<em>B<\/em>,\u227a) as a subset of <strong>RI<\/strong>(<em><em>B<\/em><\/em>,\u2aaf).  That is easy: an ideal <em>I&#8217;<\/em> of <strong>RI<\/strong>(<em><em>B<\/em><\/em>,\u2aaf) is in <strong>RI<\/strong>(<em>B<\/em>,\u227a) if and only if it is rounded with respect to \u227a.  I claim that this is equivalent to the following formula:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(R) for every <em>x<\/em> \u2208 <em>B<\/em>, if <em>x<\/em> \u2208 <em>I&#8217;<\/em> then there is a <em>y<\/em> \u2208 <em>B<\/em> such that <em>x<\/em>\u227a<em>y<\/em> and <em>y<\/em> \u2208 <em>I&#8217;<\/em>.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Let us check the equivalence.  Every rounded ideal <em>I<\/em> of <em>B<\/em>,\u227a satisfies (R): this is the special case <em>n<\/em>=1 of the definition of directedness for \u227a.  Conversely, if <em>I&#8217;<\/em> is an ideal of <em><em>B<\/em><\/em>,\u2aaf that satisfies (R), then for every finite collection of points <em>x<\/em><sub>1<\/sub>, &#8230;, <em>x<sub>n<\/sub><\/em> in <em>I&#8217;<\/em>, since <em>I&#8217;<\/em> is an ideal of <em><em>B<\/em><\/em>,\u2aaf, there is a point <em>x&#8217;<\/em> of <em>I&#8217;<\/em> such that <em>x<\/em><sub>1<\/sub>, &#8230;, <em>x<sub>n<\/sub><\/em> \u2aaf <em>x&#8217;<\/em>.  We now use (R) and find a point <em>y<\/em> in <em>I&#8217;<\/em> such that <em>x&#8217;<\/em> \u227a <em>y<\/em>, whence <em>x<\/em><sub>1<\/sub>, &#8230;, <em>x<sub>n<\/sub><\/em> \u227a <em>y<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The formula (R) can be restated by noticing that atomic subformulae such as &#8221; <em>x<\/em> \u2208 <em>I&#8217;<\/em> &#8221; are equivalent to &#8221; <em>I&#8217;<\/em> \u2208 [<em>x<\/em>]&#8217; &#8220;.  When <em>U<\/em> and <em>V<\/em> are open sets, we have already used the notation <em>U<\/em> \u21d2 <em>V<\/em> in the past to denote the set of elements that are in <em>V<\/em> or not in <em>U<\/em> (i.e., such that, if they are in <em>U<\/em>, then they are in <em>V<\/em>), and we have called them UCO subsets, following R. Heckmann: see <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/wp-admin\/post.php?post=1737\">this post<\/a> on countably presented locales or <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/wp-admin\/post.php?post=1998\">that post<\/a> on sober subspaces and the Skula topology, for example.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence an ideal <em>I&#8217;<\/em> of <em><em>B<\/em><\/em>,\u2aaf is in <strong>RI<\/strong>(<em>B<\/em>,\u227a) if and only if it is in all the UCO subsets [<em>x<\/em>]&#8217; \u21d2 \u222a {[<em>y<\/em>]&#8217; | <em>y<\/em> \u2208 <em>B<\/em> such that <em>x<\/em>\u227a<em>y<\/em>}, where <em>x<\/em> ranges over the elements of <em>B<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Just before the section on &#8220;The Skula topology&#8221; in the <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/wp-admin\/post.php?post=1998\">second post<\/a> I cited above, I said that any intersection of UCO sets in a sober space is sober.  And certainly, <strong>RI<\/strong>(<em><em>B<\/em><\/em>,\u2aaf) is sober, since any continuous dcpo is sober in its Scott topology (see Proposition 8.2.12 (b) in the <a href=\"https:\/\/www.cambridge.org\/gb\/knowledge\/isbn\/item7069109\/Non-Hausdorff%20Topology%20and%20Domain%20Theory\/?site_locale=en_GB\">book<\/a>), and its Scott topology coincides with the weak topology (Lemma A).  This allows us to obtain the following result easily (although not by elementary means).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Lemma B.<\/strong> For every set <em>B<\/em>, for every transitive relation \u227a on <em>B<\/em>, <strong>RI<\/strong>(<em>B<\/em>,\u227a) is a sober space in the weak topology.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">When <em>B<\/em> is countable, <strong>RI<\/strong>(<em>B<\/em>,\u227a) is not just an intersection of UCO subsets, but a <em>countable<\/em> intersection of UCO subsets, namely a so-called <strong>\u03a0<\/strong><sup>0<\/sup><sub>2<\/sub> subspace (the definition, at last!).  In that case, also, <strong>RI<\/strong>(<em><em>B<\/em><\/em>,\u2aaf) is also not just a continuous dcpo, but an \u03c9-continuous dcpo: in other words, it has a countable basis, isomorphic to <em>B<\/em>.  But every \u03c9-continuous dcpo is a quasi-Polish space with its Scott topology [2, Corollary 45], and quasi-Polishness is preserved by taking <strong>\u03a0<\/strong><sup>0<\/sup><sub>2<\/sub> subspaces [2, Corollary 23].  Hence we obtain one half of the promised result by M. de Brecht.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Proposition C.<\/strong>  For every countable set <em>B<\/em>, for every transitive relation \u227a on <em>B<\/em>, <strong>RI<\/strong>(<em>B<\/em>,\u227a) is a quasi-Polish space in the weak topology.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">One may wonder whether we could embed <em>B<\/em>, with a suitable topology, inside <strong>RI<\/strong>(<em>B<\/em>,\u227a), just as we can do for an abstract basis <em>B<\/em>,\u227a, but now without interpolation.  I do not think you can.  For one, we will now prove a converse to Proposition C, and <em>none<\/em> of the elements of <em>B<\/em> we will obtain will come from the quasi-Polish space we will be given.  That will be essential.  Another reason is sketched in the Appendix at the end of this post.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Quasi-Polish spaces as rounded ideals<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">In order to show the promised converse of Proposition C, we need to have a look back at <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=838\">quasi-ideal domains<\/a>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A <em>quasi-ideal domain<\/em> is any algebraic dcpo in which every element below a finite element is finite.  The other elements are called <em>limit<\/em> elements, and the definition means that you have a lower stratum of finite elements, and an upper stratum of limit elements, and they do not mix: no limit element is below any finite element.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In the <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/wp-admin\/post.php?post=838\">Ideal Domains III post<\/a>, I have shown that every continuous Yoneda-complete quasi-metric space <em>X<\/em> embeds into an algebraic dcpo, and in fact, in a very specific way: as the subspace of limit elements of a quasi-ideal domain.  The construction shows that every \u03c9-continuous Yoneda-complete quasi-metric space <em>X<\/em> (in particular, every quasi-Polish space) embeds as the subspace of limit elements of a quasi-ideal domain with countably many finite elements.  This appears explicitly as Theorem 8.18 of [4].<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence, let <em>X<\/em> be a quasi-Polish space.  Up to isomorphism, we can equate <em>X<\/em> with the set <em>Y<\/em>\u2013<em>B<\/em> of limit elements of some quasi-ideal domain <em>Y<\/em>, with a countable set <em>B<\/em> of finite elements.  Let \u2264 be the ordering on <em>Y<\/em>, and &lt; be its strict part, namely <em>x<\/em>&lt;<em>y<\/em> if and only if <em>x<\/em>\u2264<em>y<\/em> and <em>x<\/em>\u2260<em>y<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In general, that is, ignoring the countability of <em>B<\/em>, we have the following.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Proposition D.<\/strong>  Let <em>Y<\/em> be a quasi-ideal domain, with set <em>B<\/em> of finite elements.  Equip <em>Y<\/em> with the Scott topology, and let <em>X<\/em> be <em>Y<\/em>\u2013<em>B<\/em> with the subspace topology.  Let also &lt; be the strict part of the ordering \u2264 on <em>Y<\/em>.  Then <em>X<\/em> and <strong>RI<\/strong>(<em>B<\/em>,&lt;) are homeomorphic.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Proof.<\/em>  Let us examine the shape of rounded ideals <em>I<\/em> of <em>B<\/em>,&lt;.  <em>I<\/em> is non-empty, so let us pick <em>b<\/em><sub>1<\/sub> in <em>I<\/em>.  By the <em>n<\/em>=0 case of directedness, there is an element <em>b<\/em><sub>2<\/sub> in <em>I<\/em> such that <em>b<\/em><sub>1<\/sub>&lt;<em>b<\/em><sub>2<\/sub>.  Similarly, there is an element <em>b<\/em><sub>3<\/sub> in <em>I<\/em> such that <em>b<\/em><sub>2<\/sub>&lt;<em>b<\/em><sub>3<\/sub>, and so on.  This allows us to build an infinite, strictly increasing sequence <em>b<\/em><sub>1<\/sub> &lt; &#8230; &lt; <em>b<sub>n<\/sub><\/em> &lt; &#8230; of elements of <em>I<\/em>.  Let <em>x<\/em> be the supremum of that sequence in <em>Y<\/em>.  If <em>x<\/em> were a finite element of <em>Y<\/em>, then we would have <em>x<\/em>\u2264<em>b<\/em><sub><em>i<\/em><\/sub> for some index <em>i<\/em>, whence <em>x<\/em> &lt; <em>b<\/em><sub><em>i<\/em>+1<\/sub> \u2264 <em>x<\/em>, which is impossible.  Therefore <em>x<\/em> is a limit element, in <em>X<\/em>.  Also, <em>x<\/em> \u2264 sup <em>I<\/em>, since <em>x<\/em> is obtained as the supremum of a subset of <em>I<\/em>.  Since <em>Y<\/em> is a quasi-ideal domain, sup <em>I<\/em> must also be in <em>X<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This defines a function sup : <strong>RI<\/strong>(<em>B<\/em>,&lt;) \u2192 <em>X<\/em>, which maps every rounded ideal of <em>B<\/em> to its supremum, which lies in <em>X<\/em>, as we have just seen.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let us verify that this function is continuous.  The topology of <em>X<\/em> has a base of sets of the form \u2191<em>b<\/em> \u2229 <em>X<\/em>, <em>b<\/em> \u2208 <em>B<\/em>.  Then sup<sup>-1<\/sup>(\u2191<em>b<\/em> \u2229 <em>X<\/em>) is the set of rounded ideals <em>I<\/em> of <em>B<\/em>,&lt; such that <em>b<\/em>\u2264sup <em>I<\/em>, or equivalently such that <em>b<\/em> \u2208 <em>I<\/em>, since <em>b<\/em> is a finite element of <em>Y<\/em>.  In other words, sup<sup>-1<\/sup>(\u2191<em>b<\/em> \u2229 <em>X<\/em>) = [<em>b<\/em>].<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Since every subbasic open set [<em>b<\/em>] of <strong>RI<\/strong>(<em>B<\/em>,&lt;) is the inverse image of some open subset (namely \u2191<em>b<\/em> \u2229 <em>X<\/em>) of <em>X<\/em>, the function sup is almost open, too.  The argument is as for the function <em>i<\/em>, dealt with earlier.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">It is also clear that sup is surjective.  Every point <em>x<\/em> of <em>X<\/em> is by definition the supremum of the directed (with respect to \u2264) family <em>D<sub>x<\/sub><\/em> of points <em>b<\/em> \u2208 <em>B<\/em> below <em>x<\/em>.  <em>D<sub>x<\/sub><\/em> is an ideal of <em>B<\/em>,\u2264, and we claim that it is a rounded ideal of <em>B<\/em>,&lt;.  To this end, we have already noticed that it was enough to check that <em>D<sub>x<\/sub><\/em> satisfies formula (R), namely: for every <em>b<\/em> in <em>D<sub>x<\/sub><\/em>, there is an element <em>b&#8217;<\/em> in <em>D<sub>x<\/sub><\/em> such that <em>b<strong>&lt;<\/strong>b&#8217;<\/em>.  This is easy.  We assume, on the contrary that <em>D<sub>x<\/sub><\/em> has an element <em>b<\/em> such that <em>b<strong>&lt;<\/strong>b&#8217;<\/em> for no <em>b&#8217;<\/em> in <em>D<sub>x<\/sub><\/em>; in other words, that <em>D<sub>x<\/sub><\/em> has a maximal element <em>b<\/em>.  Since <em>D<sub>x<\/sub><\/em> is directed with respect to \u2264, <em>b<\/em> would be the largest element of <em>D<sub>x<\/sub><\/em>, so that <em>b<\/em>=sup <em>D<sub>x<\/sub><\/em>=<em>x<\/em>, which is impossible since <em>b<\/em> is finite and <em>x<\/em> is not.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Finally, we claim that sup is injective.  Let <em>I<\/em> be a rounded ideal of <em>B<\/em>,&lt;, <em>x<\/em> be its supremum.  We only have to show that <em>I<\/em> is equal to <em>D<sub>x<\/sub><\/em>.  The inclusion <em>I<\/em> \u2286 <em>D<sub>x<\/sub><\/em> is obvious, so it suffices to show that <em>D<sub>x<\/sub><\/em> \u2286 <em>I<\/em>.  For every <em>b<\/em> in <em>D<sub>x<\/sub><\/em>, <em>b<\/em> \u2264 <em>x<\/em> = sup <em>I<\/em>, and since <em>b<\/em> is finite, there is an element <em>b&#8217;<\/em> of <em>I<\/em> such that <em>b<\/em>\u2264<em>b&#8217;<\/em>.  If <em>b<\/em>=<em>b&#8217;<\/em>, then <em>b<\/em> is in <em>I<\/em>.  Otherwise, <em>b<\/em>&lt;<em>b&#8217;<\/em>, and since <em>I<\/em> is downwards-closed with respect to &lt;, <em>b<\/em> is again in <em>I<\/em>.  \u2610<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In summary, sup is a homeomorphism.  As we have said above, every quasi-Polish space <em>X<\/em> is representable, up to homeomorphism, as the set <em>X<\/em>=<em>Y<\/em>\u2013<em>B<\/em> of limit elements of some quasi-ideal domain <em>Y<\/em> with a countable set <em>B<\/em> of finite elements.  Together with Proposition C, Proposition D therefore implies the following.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Theorem (de Brecht, [1]).<\/strong>  The rounded ideal completions <strong>RI<\/strong>(<em>B<\/em>,\u227a) of countable sets <em>B<\/em> equipped with a transitive relation \u227a (with the weak topology) are exactly the quasi-Polish spaces, up to homeomorphism.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Open questions<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Naturally, this begs the question of what the rounded ideal completions of not necessarily countable sets <em>B<\/em>, with a transitive relation \u227a, can be.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>We know (Lemma A) that those spaces are all sober.  Are all sober spaces representable as <strong>RI<\/strong>(<em>B<\/em>,\u227a) up to homeomorphism for some <em>B<\/em>,\u227a with \u227a transitive?  I tend to doubt it.<\/li>\n\n\n\n<li>We also know that all the continuous Yoneda-complete quasi-metric spaces, with their <em>d<\/em>-Scott topology, have a quasi-ideal domain (see <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/wp-admin\/post.php?post=838\">Ideal Domains III<\/a>, or Section 8 of [4]).  Hence, by Proposition D, they can all be represented as <strong>RI<\/strong>(<em>B<\/em>,\u227a) up to homeomorphism for some <em>B<\/em>,\u227a with \u227a transitive.  Are those the only such spaces?  In other words, can we always equip any rounded ideal completion <strong>RI<\/strong>(<em>B<\/em>,\u227a) (where \u227a is transitive) with a quasi-metric <em>d<\/em> that makes it Yoneda-complete, and such that the resulting <em>d<\/em>-Scott topology coincides with the weak topology?  I also doubt it.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Appendix: you (probably) cannot embed <em>B<\/em> into <strong>RI<\/strong>(<em>B<\/em>,\u227a) without interpolation<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">I said that I do not think you can embed <em>B<\/em> into <strong>RI<\/strong>(<em>B<\/em>,\u227a) when \u227a is not interpolative.  Here is another reason.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Jimmie Lawson (see [3], or Exercise 8.2.48 in the <a href=\"https:\/\/www.cambridge.org\/gb\/knowledge\/isbn\/item7069109\/Non-Hausdorff%20Topology%20and%20Domain%20Theory\/?site_locale=en_GB\">book<\/a>) showed that, for any abstract basis <em>B<\/em>,\u227a, <strong>RI<\/strong>(<em>B<\/em>,\u227a) is the sobrification <strong>S<\/strong>(<em>B<\/em>) of <em>B<\/em>.  Here <em>B<\/em> is given the <em>pseudoScott<\/em> topology, whose subbasic open sets are the sets \u21d1<em>x<\/em> = {<em>y<\/em> \u2208 <em>B<\/em> | <em>x<\/em>\u227a<em>y<\/em>}.  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 use the so-called rounded topology instead, but, thanks to interpolation, the two topologies coincide, if we start from an abstract basis.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Generalizing away from this situation, and merely assuming \u227a transitive, not necessarily interpolative, I will let you verify that the function <em>c<\/em> : <strong>RI<\/strong>(<em>B<\/em>,\u227a) \u2192 <strong>S<\/strong>(<em>B<\/em>) that maps every rounded ideal <em>I<\/em> in <em>B<\/em> to its closure in <em>B<\/em> (with the pseudoScott topology; the rounded &#8216;topology&#8217; is not even a topology without interpolation) is continuous and almost open.  You should first show that the sets \u2662\u21d1<em>x<\/em> of irreducible closed subsets of <em>B<\/em> that intersect \u21d1<em>x<\/em> form a base, not just a subbase of the topology on <strong>S<\/strong>(<em>B<\/em>); then, that <em>c<\/em><sup>-1<\/sup>(\u2662\u21d1<em>x<\/em>)=[<em>x<\/em>] for every <em>x<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The function <em>c<\/em> is also injective, so <em>c<\/em> is a topological embedding.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Now imagine that there were an embedding \u03b5 of <em>B<\/em> into <strong>RI<\/strong>(<em>B<\/em>,\u227a) which, composed with <em>c<\/em>, yielded the usual embedding \u03b7 of <em>B<\/em> into <strong>S<\/strong>(<em>B<\/em>).  Up to homeomorphism, this allows us to consider <em>B<\/em> as a subspace of <strong>RI<\/strong>(<em>B<\/em>,\u227a), itself a subspace of <strong>S<\/strong>(<em>B<\/em>).  Since <strong>RI<\/strong>(<em>B<\/em>,\u227a) is sober (Lemma A), it would be <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/wp-admin\/post.php?post=1998\">Skula-closed<\/a> in the sober space <strong>S<\/strong>(<em>B<\/em>), and <strong>S<\/strong>(<em>B<\/em>) would be the <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/wp-admin\/post.php?post=1998\">Skula-closure<\/a> of <em>B<\/em> in <strong>S<\/strong>(<em>B<\/em>), namely the smallest Skula-closed subset containing <em>B<\/em>; that would imply that <em>c<\/em> is surjective, hence a homeomorphism.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">You can show that the inverse <em>j<\/em> of <em>c<\/em> would necessarily map every point <em>x<\/em> of <em>B<\/em> (equated with its image cl({<em>x<\/em>}) in <strong>S<\/strong>(<em>B<\/em>)) to \u21d3<em>x<\/em>.  This is by Stone duality.  The Stone dual <strong>O<\/strong>(<em>c<\/em>) : <strong>O<\/strong>(<strong>S<\/strong>(<em>B<\/em>)) \u2192 <strong>O<\/strong>(<strong>RI<\/strong>(<em>B<\/em>,\u227a)) maps \u2662\u21d1<em>y<\/em> to [<em>y<\/em>] for every point <em>y<\/em> of <em>B<\/em>, hence the Stone dual <strong>O<\/strong>(<em>j<\/em>) of <em>j<\/em> would have to map [<em>y<\/em>] to \u2662\u21d1<em>y<\/em>.  Then, for all <em>x<\/em>, <em>y<\/em> in <em>B<\/em>, <em>y<\/em> is in <em>j<\/em>(<em>x<\/em>) if and only if <em>j<\/em>(<em>x<\/em>) is in [<em>y<\/em>], if and only if cl({<em>x<\/em>}) is in \u2662\u21d1<em>y<\/em>, if and only if <em>y<\/em>\u227a<em>x<\/em>; that indeed shows that <em>j<\/em>(<em>x<\/em>) must be equal to \u21d3<em>x<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The issue is that \u21d3<em>x<\/em> is <em>not<\/em> a rounded ideal for every <em>x<\/em> in <em>B<\/em>, unless \u227a is interpolative.  Therefore, if \u227a is not interpolative, then the inverse <em>j<\/em> cannot exist; in particular, there cannot be any embedding \u03b5 of <em>B<\/em> into <strong>RI<\/strong>(<em>B<\/em>,\u227a) such that <em>c<\/em> o \u03b5 = \u03b7.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">I am leaving out a lot of details, and this is also perhaps not the most direct argument.  I am only mentioning this to stress that, without interpolation, embedding <em>B<\/em> into <strong>RI<\/strong>(<em>B<\/em>,\u227a) is quite probably hopeless; and if you succeed in doing it, it will probably be very unnatural anyway, in the sense that the embedding \u03b5 will not satisfy <em>c<\/em> o \u03b5 = \u03b7.<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Matthew de Brecht. &nbsp;Some notes on spaces of ideals and computable topology.  arXiv <a href=\"https:\/\/arxiv.org\/abs\/2004.13375\">2004.13375<\/a>, April 2020.<\/li>\n\n\n\n<li>Matthew de Brecht. &nbsp;<a href=\"https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0168007212001820\">Quasi-Polish spaces<\/a>. &nbsp;Annals of Pure and Applied Logic, <a href=\"https:\/\/www.sciencedirect.com\/science\/journal\/01680072\/164\/3\">Volume 164, Issue 3<\/a>, March 2013, pages 356-381.<\/li>\n\n\n\n<li>Jimmie D. Lawson.  The round ideal completion via sobrification.  Topology Proceedings, volume 22, 1997, pages 261-274.<\/li>\n\n\n\n<li>Jean Goubault-Larrecq and Kok Min Ng.  <a href=\"https:\/\/arxiv.org\/pdf\/1606.05445.pdf\">A few notes on formal balls<\/a>.  <a href=\"https:\/\/lmcs.episciences.org\">Logical Methods in Computer Science<\/a>, <a href=\"https:\/\/doi.org\/10.23638\/LMCS-13(4:18)2017\">13(4:18)<\/a>, pages 1-34, November 2017. Special Issue of the Domains XII Workshop.&nbsp;<\/li>\n<\/ol>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/wp-content\/uploads\/2016\/08\/jgl-2011.png\" alt=\"jgl-2011\" class=\"wp-image-993\" width=\"60\" height=\"83\"\/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">\u2014 <a href=\"https:\/\/www.lsv.ens-paris-saclay.fr\/~goubault\/?l=en\">Jean Goubault-Larrecq<\/a> (May 20th, 2020)<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The rounded ideal completion RI(B,\u227a) of a transitive, interpolative relation \u227a on a set B (an abstract basis), is always a continuous dcpo, and every continuous dcpo can be written this way, up to isomorphism. Quite some time ago, Matthew &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=2416\">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":{"_crdt_document":"","footnotes":""},"class_list":["post-2416","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/2416","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=2416"}],"version-history":[{"count":73,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/2416\/revisions"}],"predecessor-version":[{"id":5901,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/2416\/revisions\/5901"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2416"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}