{"id":3081,"date":"2021-01-23T12:04:24","date_gmt":"2021-01-23T11:04:24","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=3081"},"modified":"2022-11-19T14:59:53","modified_gmt":"2022-11-19T13:59:53","slug":"quasi-uniform-spaces-iv-formal-balls-a-proposal","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=3081","title":{"rendered":"Quasi-uniform spaces IV: Formal balls\u2014a proposal"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Formal balls are an extraordinarily useful notion in the study of quasi-metric, and even hemi-metric spaces: see Section 7.3 of the <a href=\"https:\/\/www.cambridge.org\/gb\/knowledge\/isbn\/item7069109\/Non-Hausdorff%20Topology%20and%20Domain%20Theory\/?site_locale=en_GB\">book<\/a>, and [1].  Is there any way of extending the notion to the case of quasi-uniform spaces?  This is what I would like to start investigating in this post.  In particular, this post is pretty experimental, and I don&#8217;t make any guarantee that any of what I am going to say leads to anything of any interest whatsoever.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let me remind you of what formal balls are and a bit of what you can do with them.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Given a hemi-metric space <em>X<\/em>,<em>d<\/em>, its space <strong>B<\/strong>(<em>X<\/em>,<em>d<\/em>) of formal balls is the set of pairs (formal balls) of the form (<em>x<\/em>,<em>r<\/em>) where <em>x<\/em> is a point in <em>X<\/em> and <em>r<\/em> is a non-negative real.<\/li>\n\n\n\n<li>Formal balls are preordered (and, in fact, ordered if <em>d<\/em> is a quasi-metric, not just a hemi-metric) by (<em>x,r<\/em>) \u2264<sup><em>d<\/em>+<\/sup> (<em>y<\/em>,<em>s<\/em>) if and only if <em>d<\/em>(<em>x<\/em>,<em>y<\/em>) \u2264 <em>r<\/em>\u2013<em>s<\/em>.<\/li>\n\n\n\n<li>The Kostanek-Waszkiewicz theorem (Theorem 7.4.27 in the <a href=\"https:\/\/www.cambridge.org\/gb\/knowledge\/isbn\/item7069109\/Non-Hausdorff%20Topology%20and%20Domain%20Theory\/?site_locale=en_GB\">book<\/a>) states that <strong>B<\/strong>(<em>X<\/em>,<em>d<\/em>) is a dcpo if and only if <em>X<\/em>,<em>d<\/em> is Yoneda-complete.<\/li>\n\n\n\n<li>The Romaguera-Valero theorem (Theorem 7.3.11) states that <em>X<\/em>,<em>d<\/em> is Smyth-complete if and only if <strong>B<\/strong>(<em>X<\/em>,<em>d<\/em>) is a continuous dcpo with \u227a<sup><em>d<\/em>+<\/sup> as way-below relation, where (<em>x<\/em>,<em>r<\/em>) \u227a<sup><em>d<\/em>+<\/sup> (<em>y<\/em>,<em>s<\/em>) is defined by <em>d<\/em>(<em>x<\/em>,<em>y<\/em>) &lt; <em>r<\/em>\u2013<em>s<\/em>.<\/li>\n\n\n\n<li>A deeper understanding of the latter is obtained by realizing that <strong>B<\/strong>(<em>X<\/em>,<em>d<\/em>) itself has a hemi-metric <em>d<\/em><sup>+<\/sup>, defined by <em>d<\/em><sup>+<\/sup>((<em>x<\/em>,<em>r<\/em>), (<em>y<\/em>,<em>s<\/em>)) \u225d max(<em>d<\/em>(<em>x<\/em>,<em>y<\/em>)\u2013<em>r<\/em>+<em>s<\/em>, 0), that <strong>B<\/strong>(<em>X<\/em>,<em>d<\/em>) is a c-space in the open ball topology of <em>d<\/em><sup>+<\/sup>, and that \u227a<sup><em>d<\/em>+<\/sup> is the relation we would expect to be of some interest in a c-space, namely: (<em>x<\/em>,<em>r<\/em>) \u227a<sup><em>d<\/em>+<\/sup> (<em>y<\/em>,<em>s<\/em>) if and only if (<em>y<\/em>,<em>s<\/em>) is in the interior of the upward closure of (<em>x<\/em>,<em>r<\/em>).  Then <em>X<\/em>,<em>d<\/em> is Smyth-complete if and only if <strong>B<\/strong>(<em>X<\/em>,<em>d<\/em>) is sober in the open ball topology of <em>d<\/em><sup>+<\/sup> (Theorem 8.3.40).<\/li>\n\n\n\n<li>One can build a completion <strong>S<\/strong>(<em>X<\/em>,<em>d<\/em>), <em>d<\/em><sup>+<\/sup><strong><em><sub>H<\/sub><\/em><\/strong> as follows (section 7.5, see also [2]).  Its elements are rounded ideals of formal balls, built from the abstract basis <strong>B<\/strong>(<em>X<\/em>,<em>d<\/em>), \u227a<sup><em>d<\/em>+<\/sup>.  This is an abstract basis because c-spaces and abstract bases are essentially the same thing (Exercise 8.3.46).  And the quasi-metric <em>d<\/em><sup>+<\/sup><strong><em><sub>H<\/sub><\/em><\/strong> is the Hausdorff-Hoare quasi-metric, defined by  <em>d<\/em><sup>+<\/sup><strong><em><sub>H<\/sub><\/em><\/strong>(<em>D<\/em>, <em>D&#8217;<\/em>) \u225d sup<sub>(<em>x<\/em>,<em>r<\/em>) \u2208 <em>D<\/em><\/sub> inf<sub>(<em>y<\/em>,<em>s<\/em>) \u2208 <em>D<\/em>&#8216;<\/sub> <em>d<\/em><sup>+<\/sup>((<em>x<\/em>,<em>r<\/em>), (<em>y<\/em>,<em>s<\/em>)).<\/li>\n\n\n\n<li><strong>S<\/strong>(<em>X<\/em>,<em>d<\/em>), <em>d<\/em><sup>+<\/sup><strong><em><sub>H<\/sub><\/em><\/strong> is always Yoneda-complete, and in fact algebraic Yoneda-complete (Exercise 7.5.11).<\/li>\n\n\n\n<li><strong>S<\/strong>(<em>X<\/em>,<em>d<\/em>), <em>d<\/em><sup>+<\/sup><strong><em><sub>H<\/sub><\/em><\/strong> is the free Yoneda-complete quasi-metric space above <em>X<\/em>,<em>d<\/em>, in a precise sense (Fact 7.5.23).  This notably means that there is a 1-Lipschitz map \u03b7<strong><sup>S<\/sup><\/strong> : <em>X<\/em>,<em>d<\/em> \u2192 <strong>S<\/strong>(<em>X<\/em>,<em>d<\/em>), <em>d<\/em><sup>+<\/sup><strong><em><sub>H<\/sub><\/em><\/strong>, called the <em>unit<\/em> (explicitly, \u03b7<strong><sup>S<\/sup><\/strong>(<em>x<\/em>) is the set of all formal balls \u227a<sup><em>d<\/em>+<\/sup> (<em>x<\/em>,0)), and that every 1-Lipschitz map <em>f<\/em> : <em>X<\/em>,<em>d<\/em> \u2192 <em>Y<\/em>,<em>d&#8217;<\/em> to a Yoneda-complete quasi-metric space <em>Y<\/em>,<em>d&#8217;<\/em> extends uniquely to a 1-Lipschitz <em>continuous<\/em> map <em>f<\/em>\u2020 : <strong>S<\/strong>(<em>X<\/em>,<em>d<\/em>), <em>d<\/em><sup>+<\/sup><strong><em><sub>H<\/sub><\/em><\/strong> \u2192 <em>Y<\/em>,<em>d&#8217;<\/em>; and by extending I mean that <em>f<\/em>\u2020 o \u03b7<strong><sup>S<\/sup><\/strong> = <em>f<\/em> (Propositoin 7.5.22).<\/li>\n\n\n\n<li><strong>S<\/strong>(<em>X<\/em>,<em>d<\/em>), <em>d<\/em><sup>+<\/sup><strong><em><sub>H<\/sub><\/em><\/strong> is isomorphic to <em>X<\/em>,<em>d<\/em> through the unit \u03b7<strong><sup>S<\/sup><\/strong> if and only if <em>X<\/em>,<em>d<\/em> is Smyth-complete, if and ony if \u03b7<strong><sup>S<\/sup><\/strong> is bijective (Proposition 7.5.15).<\/li>\n\n\n\n<li><strong>S<\/strong>(<em>X<\/em>,<em>d<\/em>), <em>d<\/em><sup>+<\/sup><strong><em><sub>H<\/sub><\/em><\/strong> coincides with the usual Cauchy completion when <em>d<\/em> is a metric, up to natural isometry (Exercise 7.5.21).<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">I will suggest a notion of formal ball for quasi-uniform spaces.  Not everything will run as smoothly as in the hemi-metric case, but let me say that this is promising so far.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In the sequel, we fix a quasi-uniform space <em>X<\/em>, with quasi-uniformity <strong><em>U<\/em><\/strong>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">A proposal for formal balls in the quasi-uniform setting<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Given a hemi-metric space <em>X<\/em>, <em>d<\/em>, a formal ball is a pair (<em>x<\/em>,<em>r<\/em>) where <em>x<\/em> is a point of <em>X<\/em> and <em>r<\/em> is a non-negative real number. Up to some degree of approximation, we may encode <em>r<\/em> by the set [\u2264<em>r<\/em>] \u225d {(<em>x<\/em>,<em>y<\/em>) | <em>d<\/em>(<em>x<\/em>,<em>y<\/em>)\u2264<em>r<\/em>}.  This is an entourage, provided that <em>r<\/em>&gt;0.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let me define quasi-uniform formal balls by analogy, as pairs (<em>x<\/em>, <em>R<\/em>) where <em>x<\/em> is a point of <em>X<\/em> and <em>R<\/em> is an entourage, in <strong><em>U<\/em><\/strong>. Of course, the entourage <em>R<\/em> plays the r\u00f4le of a radius here. Let <strong>B<\/strong><sub>0<\/sub>(<em>X<\/em>,<strong><em>U<\/em><\/strong>) denote the set of all quasi-uniform formal balls on <em>X<\/em>, with quasi-unifomity <strong><em>U<\/em><\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">There is one thing we are missing with this proposal: there is no analogue of the formal balls (<em>x<\/em>,<em>r<\/em>) with <em>r<\/em>=0.  Indeed, [\u22640] is <em>not<\/em> generally an entourage.  This is a problem inasmuch as this prevents us from embedding <em>X<\/em> into <strong>B<\/strong><sub>0<\/sub>(<em>X<\/em>,<strong><em>U<\/em><\/strong>), as we could do it in the hemi-metric case, through the map <em>x<\/em> \u21a6 (<em>x<\/em>,0).  We may fix this by taking the disjoint union of <strong>B<\/strong><sub>0<\/sub>(<em>X<\/em>,<strong><em>U<\/em><\/strong>) with <em>X<\/em> itself, for example, but I will not do this.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">There is a further issue with taking entourages as an extended form of radii.  The closest analogue of addition of radii is relation composition, since [\u2264<em>r<\/em>] o [\u2264<em>s<\/em>] \u2286 [\u2264<em>r<\/em>+<em>s<\/em>], but this is not a perfect analogy, since we only have an inclusion here, not an equality. Worse, that &#8216;addition&#8217; obtained as relation composition is <em>not commutative<\/em>, and not invertible.  This is a source of constant trouble&#8230; but we will learn to live with it.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">All right, how do we order our new kind of formal balls?  I must say I have had several ideas, but only one seems to give any kind of decent results.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">The preordering \u2264<strong><em><sup>U+<\/sup><\/em><\/strong> on formal balls<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">We <em>pre<\/em>order formal balls by: (<em>x<\/em>,<em>R<\/em>) \u2264<strong><em><sup>U+<\/sup><\/em><\/strong> (<em>y<\/em>,<em>S<\/em>) if and only if <em>R<\/em> \u2287 <em>S<\/em> and <em>R<\/em>[<em>x<\/em>] \u2287 <em>S<\/em>[<em>y<\/em>].  Beware that this is a preordering, not an ordering in general.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">That looks like this comes from nowhere; but look carefully at the proof of the proposition, seen <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=2998\" data-type=\"page\" data-id=\"2998\">last time<\/a>, that every Smyth-complete space <em>X<\/em> is quasi-sober in its induced topology: this is exactly the relation that we had used there.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let us compare this relation \u2264<strong><em><sup>U+<\/sup><\/em><\/strong> on quasi-uniform formal balls with the usual ordering on formal balls we had in the quasi-metric case (Fact 7.3.2 in the <a href=\"https:\/\/www.cambridge.org\/gb\/knowledge\/isbn\/item7069109\/Non-Hausdorff%20Topology%20and%20Domain%20Theory\/?site_locale=en_GB\">book<\/a>): in the quasi-metric setting, (<em>x<\/em>,<em>r<\/em>) is below (<em>y<\/em>,<em>s<\/em>) if and only if <em>d<\/em>(<em>x<\/em>,<em>y<\/em>)\u2264<em>r<\/em>\u2013<em>s<\/em>.  That implies that the open ball of radius <em>r<\/em> centered at <em>x<\/em> contains the open ball of radius <em>s<\/em> centers at <em>y<\/em>, namely that <em>R<\/em>[<em>x<\/em>] \u2287 <em>S<\/em>[<em>y<\/em>], where <em>R<\/em> \u225d [\u2264<em>r<\/em>] and <em>S<\/em> \u225d [\u2264<em>s<\/em>], but is not equivalent to it.  That also implies <em>r<\/em>\u2265<em>s<\/em>, which implies <em>R<\/em> \u2287 <em>S<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence \u2264<strong><em><sup>U+<\/sup><\/em><\/strong> is somewhat close to the usual ordering on quasi-metric formal balls.  However, as I mentioned above, \u2264<strong><em><sup>U+<\/sup><\/em><\/strong> is certainly not an ordering in general, which sets it apart from the usual ordering \u2264<em><sup>d<\/sup><\/em><strong><em><sup>+<\/sup><\/em><\/strong> we have on formal balls.  In fact, for every <em>R<\/em> \u2208 <strong><em>U<\/em><\/strong><sub>0<\/sub>, for any two points <em>x<\/em> and <em>y<\/em> such that <em>R<\/em>[<em>x<\/em>]=<em>R<\/em>[<em>y<\/em>], we will have both (<em>x<\/em>,<em>R<\/em>) \u2264<strong><em><sup>U+<\/sup><\/em><\/strong> (<em>y<\/em>,<em>R<\/em>) and (<em>y<\/em>,<em>R<\/em>) \u2264<strong><em><sup>U+<\/sup><\/em><\/strong> (<em>x<\/em>,<em>R<\/em>), even when <em>x<\/em>\u2260<em>y<\/em>.  That happens even when the induced topology of <strong><em>U<\/em><\/strong> is T<sub>0<\/sub>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">The abstract basis of formal balls<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The space of formal balls of a hemi-metric space is an abstract basis (Lemma 7.3.10 in the <a href=\"https:\/\/www.cambridge.org\/gb\/knowledge\/isbn\/item7069109\/Non-Hausdorff%20Topology%20and%20Domain%20Theory\/?site_locale=en_GB\">book<\/a>), where (<em>x<\/em>,<em>r<\/em>) \u227a (<em>y<\/em>,<em>s<\/em>) if and only if <em>d<\/em>(<em>x<\/em>,<em>y<\/em>) &lt; <em>r<\/em>\u2013<em>s<\/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>, this is deduced from the fact that it is a c-space, one given the open ball topology of a new hemi-metric <em>d<\/em><sup>+<\/sup>.  None of this seems to work quite as well with quasi-uniform spaces, but we can define a structure of an abstract basis on <strong>B<\/strong><sub>0<\/sub>(<em>X<\/em>,<strong><em>U<\/em><\/strong>) directly, by analogy, as follows.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In the hemi-metric case, notice that (<em>x<\/em>,<em>r<\/em>) \u227a (<em>y<\/em>,<em>s<\/em>) if and only if there is a <em>t<\/em>&gt;0 such that (<em>x<\/em>,<em>r<\/em>) \u2264<sup><em>d<\/em>+<\/sup>(<em>y<\/em>,<em>t<\/em>+<em>s<\/em>).  By analogy, in the quasi-uniform case, we declare that (<em>x<\/em>,<em>R<\/em>) \u227a<sup><em><strong>U<\/strong><\/em>+<\/sup> (<em>y<\/em>,<em>S<\/em>) if and only if there is an entourage <em>T<\/em> \u2208 <strong><em>U<\/em><\/strong> such that (<em>x<\/em>,<em>R<\/em>) \u2264<sup><em><strong>U<\/strong><\/em>+<\/sup> (<em>y<\/em>,<em>T<\/em> o <em>S<\/em>). (Or equivalently, <em>R<\/em> \u2287 <em>T<\/em> o <em>S<\/em> and <em>R<\/em>[<em>x<\/em>] \u2287 <em>T<\/em>[<em>S<\/em>[<em>y<\/em>]].)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Note that I am using <em>T<\/em> o <em>S<\/em> as an analogue of radius addition <em>t<\/em>+<em>s<\/em>.  While it seems that I could have used <em>S<\/em> o <em>T<\/em> instead, that would definitely not work so well.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In the sequel, I will use the following often.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Fact.<\/strong>  For every two entourages <em>R<\/em> and <em>S<\/em>, <em>R<\/em> o <em>S<\/em> is again an entourage.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Indeed, since <em>R<\/em> is reflexive, <em>R<\/em> o <em>S<\/em> contains <em>S<\/em>, and any binary relation that contains an entourage is an entourage.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Lemma A.<\/strong>  The relation \u227a<sup><em><strong>U<\/strong><\/em>+<\/sup> turns <strong>B<\/strong><sub>0<\/sub>(<em>X<\/em>,<strong><em>U<\/em><\/strong>) into an abstract basis.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">First, a bit of notation, which we will reuse often. For every entourage <em>R<\/em>, there is an entourage <em>S<\/em> such that <em>S<\/em> o <em>S<\/em> \u2286 <em>R<\/em>. Let me call such an <em>S<\/em> a <em>splitter<\/em> of <em>R<\/em>. Please allow me to write \u00bd<em>R<\/em> for such a splitter: in the quasi-metric case, if <em>R<\/em>=[\u2264<em>r<\/em>], then we can take \u00bd<em>R<\/em>=<em>S<\/em> to be [\u2264\u00bd<em>r<\/em>]. There is nothing canonical in the choice of \u00bd<em>R<\/em>=<em>S<\/em>, though. However, splitters always satisfy \u00bd<em>R<\/em> o \u00bd<em>R<\/em> \u2286 <em>R<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Proof.<\/em>  We must check that \u227a<sup><em><strong>U<\/strong><\/em>+<\/sup> is transitive and interpolative (see Lemma 5.1.32 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\">Transitivity is easier.  Let us assume that (<em>x<\/em><sub>1<\/sub>,<em>R<\/em><sub>1<\/sub>) \u227a<sup><em><strong>U<\/strong><\/em>+<\/sup> (<em>x<\/em><sub>2<\/sub>,<em>R<\/em><sub>2<\/sub>) \u227a<sup><em><strong>U<\/strong><\/em>+<\/sup> (<em>x<\/em><sub>3<\/sub>,<em>R<\/em><sub>3<\/sub>).  There are two entourages <em>T<\/em><sub>1<\/sub> and <em>T<\/em><sub>2<\/sub> such that <em><em>R<\/em><\/em><sub>1<\/sub> \u2287 <em><em>T<\/em><\/em><sub>1<\/sub> o <em><em>R<\/em><\/em><sub>2<\/sub>, <em><em><em>R<\/em><\/em><\/em><sub>1<\/sub>[<em><em>x<\/em><\/em><sub>1<\/sub>] \u2287 <em><em><em>T<\/em><\/em><\/em><sub>1<\/sub>[<em><em><em>R<\/em><\/em><\/em><sub>2<\/sub>[<em><em>x<\/em><\/em><sub>2<\/sub>]], <em><em>R<\/em><\/em><sub>2<\/sub> \u2287 <em><em>T<\/em><\/em><sub>2<\/sub> o <em><em>R<\/em><\/em><sub>3<\/sub>, and <em><em>R<\/em><\/em><sub>2<\/sub>[<em><em>x<\/em><\/em><sub>2<\/sub>] \u2287 <em><em>T<\/em><\/em><sub>2<\/sub>[<em><em>R<\/em><\/em><sub>3<\/sub>[<em><em>x<\/em><\/em><sub>3<\/sub>]].  It follows that <em><em>R<\/em><\/em><sub>1<\/sub> \u2287 <em><em>T<\/em><\/em><sub>1<\/sub> o <em><em>T<\/em><\/em><sub>2<\/sub> o <em><em>R<\/em><\/em><sub>3<\/sub> and <em><em>R<\/em><\/em><sub>1<\/sub>[<em><em>x<\/em><\/em><sub>1<\/sub>] \u2287 <em><em>T<\/em><\/em><sub>1<\/sub>[<em><em><em><em>T<\/em><\/em><\/em><\/em><sub>2<\/sub>[<em><em><em><em>R<\/em><\/em><\/em><\/em><sub>3<\/sub>[<em><em><em><em>x<\/em><\/em><\/em><\/em><sub>3<\/sub>]]].  One can rewrite those as <em><em>R<\/em><\/em><sub>1<\/sub> \u2287 (<em><em>T<\/em><\/em><sub>1<\/sub> o <em><em>T<\/em><\/em><sub>2<\/sub>) o <em><em>R<\/em><\/em><sub>3<\/sub> and <em><em>R<\/em><\/em><sub>1<\/sub>[<em><em>x<\/em><\/em><sub>1<\/sub>] \u2287 (<em><em>T<\/em><\/em><sub>1<\/sub> o <em><em>T<\/em><\/em><sub>2<\/sub>)[<em><em>R<\/em><\/em><sub>3<\/sub>[<em><em>x<\/em><\/em><sub>3<\/sub>]].  Since <em><em>T<\/em><\/em><sub>1<\/sub> o <em><em>T<\/em><\/em><sub>2<\/sub> is an entourage (by the Fact above), this establishes that (<em>x<\/em><sub>1<\/sub>,<em>R<\/em><sub>1<\/sub>) \u227a<sup><em><strong>U<\/strong><\/em>+<\/sup> (<em>x<\/em><sub>3<\/sub>,<em>R<\/em><sub>3<\/sub>).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We turn to interpolation.  We assume that we have <em>n<\/em>+1 formal balls (<em>x<\/em><sub>1<\/sub>,<em>R<\/em><sub>1<\/sub>), &#8230;, (<em>x<\/em><sub><em>n<\/em><\/sub>,<em>R<\/em><sub><em>n<\/em><\/sub>) and (<em>y<\/em>,<em>S<\/em>), and that (<em>x<\/em><sub><em>i<\/em><\/sub>,<em>R<\/em><sub><em>i<\/em><\/sub>) \u227a<sup><em><strong>U<\/strong><\/em>+<\/sup> (<em>y<\/em>,<em>S<\/em>) for every <em>i<\/em>, 1\u2264<em>i<\/em>\u2264<em>n<\/em>.  Hence we have <em>n<\/em> entourages <em>T<\/em><sub>1<\/sub>, &#8230;, <em>T<\/em><sub><em>n<\/em><\/sub> such that <em><em>R<\/em><\/em><sub><em>i<\/em><\/sub> \u2287 <em><em>T<\/em><\/em><sub><em>i<\/em><\/sub> o <em><em>S<\/em><\/em> and <em><em>R<\/em><\/em><sub><em>i<\/em><\/sub>[<em><em>x<\/em><\/em><sub><em>i<\/em><\/sub>] \u2287 <em><em>T<\/em><\/em><sub><em>i<\/em><\/sub>[<em><em>S<\/em><\/em>[<em><em>y<\/em><\/em>]] for every <em>i<\/em>, 1\u2264<em>i<\/em>\u2264<em>n<\/em>.  We consider splitters \u00bd<em>T<\/em><sub><em>i<\/em><\/sub> of each of the entourages <em>T<\/em><sub><em>i<\/em><\/sub>.  Then <em>T&#8217;<\/em> \u225d \u00bd<em>T<\/em><sub>1<\/sub> \u2229 &#8230; \u2229 \u00bd<em>T<\/em><sub><em>n<\/em><\/sub> is again an entourage.  Let us define <em>S&#8217;<\/em> as <em>T&#8217;<\/em> o <em>S<\/em> (an entourage!), and let us check that (<em>y<\/em>,<em>S&#8217;<\/em>) \u227a<sup><em><strong>U<\/strong><\/em>+<\/sup> (<em>y<\/em>,<em>S<\/em>) and that (<em>x<\/em><sub><em>i<\/em><\/sub>,<em>R<\/em><sub><em>i<\/em><\/sub>) \u227a<sup><em><strong>U<\/strong><\/em>+<\/sup> (<em>y<\/em>,<em>S&#8217;<\/em>) for every <em>i<\/em>, 1\u2264<em>i<\/em>\u2264<em>n<\/em>.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(<em>y<\/em>,<em>S&#8217;<\/em>) \u227a<sup><em><strong>U<\/strong><\/em>+<\/sup> (<em>y<\/em>,<em>S<\/em>) is because <em>S&#8217;<\/em> \u2287 <em>T&#8217;<\/em> o <em>S<\/em> and <em>S&#8217;<\/em>[<em>y<\/em>] \u2287 <em>T<\/em>&#8216;[<em>S<\/em>[<em>y<\/em>]]; in fact those are equalities, not just inclusions.<\/li>\n\n\n\n<li>(<em>x<\/em><sub><em>i<\/em><\/sub>,<em>R<\/em><sub><em>i<\/em><\/sub>) \u227a<sup><em><strong>U<\/strong><\/em>+<\/sup> (<em>y<\/em>,<em>S&#8217;<\/em>) is because <em><em>R<\/em><\/em><sub><em>i<\/em><\/sub> \u2287 \u00bd<em><em>T<\/em><\/em><sub><em>i<\/em><\/sub> o <em><em>S<\/em><\/em>&#8216; and <em><em>R<\/em><\/em><sub><em>i<\/em><\/sub>[<em><em>x<\/em><\/em><sub><em>i<\/em><\/sub>] \u2287 \u00bd<em><em>T<\/em><\/em><sub><em>i<\/em><\/sub>[<em><em>S<\/em><\/em>&#8216;[<em><em>y<\/em><\/em>]].  We justify those inclusions as follows.  For the first one, \u00bd<em><em>T<\/em><\/em><sub><em>i<\/em><\/sub> o <em><em>S<\/em><\/em>&#8216; is equal to \u00bd<em><em>T<\/em><\/em><sub><em>i<\/em><\/sub> o <em>T&#8217;<\/em> o <em>S<\/em>, which is included in \u00bd<em><em>T<\/em><\/em><sub><em>i<\/em><\/sub> o \u00bd<em><em>T<\/em><\/em><sub><em>i<\/em><\/sub> o <em>S<\/em>, hence in <em><em>T<\/em><\/em><sub><em>i<\/em><\/sub> o <em>S<\/em>, and that is included in <em><em>R<\/em><\/em><sub><em>i<\/em><\/sub>.  For the second one, every point <em>z<\/em> of \u00bd<em><em>T<\/em><\/em><sub><em>i<\/em><\/sub>[<em><em>S<\/em><\/em>&#8216;[<em><em>y<\/em><\/em>]] is such that (<em>y<\/em>,<em>z<\/em>) is in \u00bd<em><em>T<\/em><\/em><sub><em>i<\/em><\/sub> o <em><em>S<\/em><\/em>&#8216;, which is included in <em><em><em><em>T<\/em><\/em><sub><em>i<\/em><\/sub><\/em><\/em> o <em><em><em>S<\/em><\/em><\/em>, as we have just seen; so <em>z<\/em> is in <em><em>T<\/em><\/em><sub><em>i<\/em><\/sub>[<em><em>S<\/em><\/em>[<em><em>y<\/em><\/em>]], and this is included in <em><em>R<\/em><\/em><sub><em>i<\/em><\/sub>[<em><em>x<\/em><\/em><sub><em>i<\/em><\/sub>].  \u2610<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">The rounded ideal completion of <strong>B<\/strong><sub>0<\/sub>(<em>X<\/em>,<strong><em>U<\/em><\/strong>)<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Once we have an abstract basis <em>B<\/em>, \u227a, taking its rounded ideal completion <strong>RI<\/strong>(<em>B<\/em>, \u227a) gives you a continuous dcpo (Proposition 5.1.33) with a basis that is essentially <em>B<\/em> itself, modulo an encoding of element <em>b<\/em> of <em>B<\/em> as rounded ideals \u21d3<em>b<\/em>, and with a way below relation defined by <em>x<\/em>\u226a<em>y<\/em> if and only if <em>x<\/em> \u2286 \u21d3<em>b<\/em> for some <em>b<\/em> in <em>y<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence we do have a continuous dcpo <strong>RI<\/strong>(<strong>B<\/strong><sub>0<\/sub>(<em>X<\/em>,<strong><em>U<\/em><\/strong>), \u227a<sup><em><strong>U<\/strong><\/em>+<\/sup>).  Its elements are the rounded ideals <em>D<\/em> of formal balls, namely the sets of formal balls such that:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><em>D<\/em> is downwards-closed: for every (<em>y<\/em>,<em>S<\/em>) in <em>D<\/em>, for every formal ball (<em>x<\/em>,<em>R<\/em>) \u227a<sup><em><strong>U<\/strong><\/em>+<\/sup> (<em>y<\/em>,<em>S<\/em>), (<em>x<\/em>,<em>R<\/em>) is in <em>D<\/em>;<\/li>\n\n\n\n<li><em>D<\/em> is directed: for every natural number <em>n<\/em> (including the cases <em>n<\/em>=0 and <em>n<\/em>=1), and for any collection of <em>n<\/em> formal balls (<em>x<\/em><sub>1<\/sub>,<em>R<\/em><sub>1<\/sub>), &#8230;, (<em>x<\/em><sub><em>n<\/em><\/sub>,<em>R<\/em><sub><em>n<\/em><\/sub>) in <em>S<\/em>, there is a further formal ball (<em>y<\/em>,<em>S<\/em>) in <em>D<\/em> such that (<em>x<\/em><sub><em>i<\/em><\/sub>,<em>R<\/em><sub><em>i<\/em><\/sub>) \u227a<sup><em><strong>U<\/strong><\/em>+<\/sup> (<em>y<\/em>,<em>S<\/em>) for every <em>i<\/em>, 1\u2264<em>i<\/em>\u2264<em>n<\/em>.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">In the hemi-metric case, we had a similar construction <strong>RI<\/strong>(<strong>B<\/strong>(<em>X<\/em>,<em>d<\/em>), \u227a), and there was a subset of it, consisting of those rounded ideals that had <em>aperture zero<\/em>.  That subset was <strong>S<\/strong>(<em>X<\/em>,<em>d<\/em>), the formal ball completion of <em>X<\/em>,<em>d<\/em>.  Let us imitate that.  The <em>aperture<\/em> of a rounded ideal <em>D<\/em> is the infimum of the radii <em>r<\/em> of those formal balls (<em>x<\/em>,<em>r<\/em>) that occur in <em>D<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">One may be tempted to define the aperture of a quasi-uniform rounded ideal as being the intersection of all the entourages <em>R<\/em> that appear in radius position in those formal balls (<em>x<\/em>,<em>R<\/em>) that occur in <em>D<\/em>, but that would lead nowhere.  That intersection may fail to be an entourage, but what is far worse is that this intersection does not have any remarkable property per se.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Instead, let me say that a rounded ideal <em>D<\/em> \u2208 <strong>RI<\/strong>(<strong>B<\/strong>(<em>X<\/em>,<em>d<\/em>), \u227a) has <em>aperture zero<\/em> if and only if for every entourage <em>R<\/em> \u2208 <strong><em>U<\/em><\/strong>, there is an element (<em>y<\/em>,<em>S<\/em>) in <em>D<\/em> such that <em>S<\/em> is included in <em>R<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Still by analogy, let <strong>S<\/strong>(<em>X<\/em>,<strong><em>U<\/em><\/strong>) be the subset of <strong>RI<\/strong>(<strong>B<\/strong>(<em>X<\/em>,<em>d<\/em>), \u227a) of those rounded ideals with aperture zero.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">The formal ball completion<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">We can turn <strong>S<\/strong>(<em>X<\/em>,<strong><em>U<\/em><\/strong>) into a quasi-uniform space\u2014one which we would like to call its formal ball completion\u2014but this is starting to get difficult.  We first observe that <em>X<\/em> (quasi-)embeds into <strong>S<\/strong>(<em>X<\/em>,<strong><em>U<\/em><\/strong>).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Lemma B.<\/strong> For every <em>y<\/em> in <em>X<\/em>, the set \u03b7(<em>y<\/em>) \u225d {(<em>x<\/em>,<em>R<\/em>) \u2208 <strong>B<\/strong><sub>0<\/sub>(<em>X<\/em>,<strong><em>U<\/em><\/strong>) | for some <em>S<\/em> \u2208 <strong><em>U<\/em><\/strong>, (<em>x<\/em>,<em>R<\/em>) \u2264<sup><em><strong>U<\/strong><\/em>+<\/sup> (<em>y<\/em>,<em>S<\/em>)} is an element of <strong>S<\/strong>(<em>X<\/em>,<strong><em>U<\/em><\/strong>).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Proof.  We must first check that \u03b7(<em>y<\/em>) is a rounded ideal.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Downward closure.  If (<em>x<\/em><sub>1<\/sub>,<em>R<\/em><sub>1<\/sub>) \u227a<sup><em><strong>U<\/strong><\/em>+<\/sup> (<em>x<\/em><sub>2<\/sub>,<em>R<\/em><sub>2<\/sub>) and (<em>x<\/em><sub>2<\/sub>,<em>R<\/em><sub>2<\/sub>) is in \u03b7(<em>y<\/em>), then there is an entourage <em>T<\/em> such that  <em><em>R<\/em><\/em><sub>1<\/sub> \u2287 <em><em>T<\/em><\/em> o <em><em>R<\/em><\/em><sub>2<\/sub>, <em><em>R<\/em><\/em><sub>1<\/sub>[<em><em>x<\/em><\/em><sub>1<\/sub>] \u2287 <em><em>T<\/em><\/em>[<em><em>R<\/em><\/em><sub>2<\/sub>[<em><em>x<\/em><\/em><sub>2<\/sub>]], and there is an entourage <em>S<\/em> such that <em><em>R<\/em><\/em><sub>2<\/sub> \u2287 <em>S<\/em> and <em>R<\/em><sub>2<\/sub>[<em><em><em>x<\/em><\/em><\/em><sub>2<\/sub>] \u2287 <em>S<\/em>[<em>y<\/em>].  Then <em><em>R<\/em><\/em><sub>1<\/sub> \u2287 <em><em>T<\/em><\/em> o <em>S<\/em> and <em><em>R<\/em><\/em><sub>1<\/sub>[<em><em>x<\/em><\/em><sub>1<\/sub>] \u2287 <em><em>T<\/em><\/em>[<em><em><em>S<\/em><\/em><\/em>[<em><em><em>y<\/em><\/em><\/em>]], so (<em>x<\/em><sub>1<\/sub>,<em>R<\/em><sub>1<\/sub>) is in \u03b7(<em>y<\/em>).<\/li>\n\n\n\n<li>Directedness.  Let (<em>x<\/em><sub>1<\/sub>,<em>R<\/em><sub>1<\/sub>), &#8230;, (<em>x<\/em><sub><em>n<\/em><\/sub>,<em>R<\/em><sub><em>n<\/em><\/sub>) be elements of \u03b7(<em>y<\/em>).  Hence there are <em>n<\/em> entourages <em>S<\/em><sub>1<\/sub>, &#8230;, <em>S<\/em><sub><em>n<\/em><\/sub> such that <em><em>R<\/em><\/em><sub><em>i<\/em><\/sub> \u2287 <em><em>S<\/em><\/em><sub><em>i<\/em><\/sub> and <em><em>R<\/em><\/em><sub><em>i<\/em><\/sub>[<em><em>x<\/em><\/em><sub><em>i<\/em><\/sub>] \u2287 <em><em>S<\/em><\/em><sub><em>i<\/em><\/sub>[<em><em>y<\/em><\/em>] for every <em>i<\/em>, 1\u2264<em>i<\/em>\u2264<em>n<\/em>. We consider splitters \u00bd<em>S<\/em><sub><em>i<\/em><\/sub> of each of the entourages <em>S<\/em><sub><em>i<\/em><\/sub>. Then <em>S&#8217;<\/em> \u225d \u00bd<em>S<\/em><sub>1<\/sub> \u2229 &#8230; \u2229 \u00bd<em>S<\/em><sub><em>n<\/em><\/sub> is again an entourage. We check that (<em>y<\/em>,<em>S&#8217;<\/em>) \u2208 \u03b7(<em>y<\/em>) and that (<em>x<\/em><sub><em>i<\/em><\/sub>,<em>R<\/em><sub><em>i<\/em><\/sub>) \u227a<sup><em><strong>U<\/strong><\/em>+<\/sup> (<em>y<\/em>,<em>S&#8217;<\/em>) for every <em>i<\/em>, 1\u2264<em>i<\/em>\u2264<em>n<\/em>.\n<ul class=\"wp-block-list\">\n<li>(<em>y<\/em>,<em>S&#8217;<\/em>) \u2208 \u03b7(<em>y<\/em>): this boils down to the existence of an entourage <em>S<\/em> such that (<em>y<\/em>,<em>S&#8217;<\/em>) \u2264<sup><em><strong>U<\/strong><\/em>+<\/sup> (<em>y<\/em>,<em>S<\/em>); we simply take <em>S<\/em> \u225d <em>S&#8217;<\/em>.<\/li>\n\n\n\n<li>(<em>x<\/em><sub><em>i<\/em><\/sub>,<em>R<\/em><sub><em>i<\/em><\/sub>) \u227a<sup><em><strong>U<\/strong><\/em>+<\/sup> (<em>y<\/em>,<em>S&#8217;<\/em>): we check that <em>R<\/em><sub><em>i<\/em><\/sub> \u2287 <em>S&#8217;<\/em> o <em>S&#8217;<\/em> and that <em><em>R<\/em><\/em><sub><em>i<\/em><\/sub>[<em><em>x<\/em><\/em><sub><em>i<\/em><\/sub>] \u2287 <em><em>S<\/em>&#8216;<\/em>[<em>S&#8217;<\/em>[<em><em>y<\/em><\/em>]], using the fact that <em>S&#8217;<\/em> \u2286 \u00bd<em>S<\/em><sub><em>i<\/em><\/sub> and that \u00bd<em>S<\/em><sub><em>i<\/em><\/sub> o \u00bd<em>S<\/em><sub><em>i<\/em><\/sub> \u2286 <em>S<\/em><sub><em>i<\/em><\/sub>.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Next, we must check that \u03b7(<em>y<\/em>) has aperture zero.  For every entourage <em>R<\/em> in <strong><em>U<\/em><\/strong>, (<em>y<\/em>,<em>R<\/em>) is in \u03b7(<em>y<\/em>): indeed (<em>y<\/em>,<em>R<\/em>) \u2264<sup><em><strong>U<\/strong><\/em>+<\/sup> (<em>y<\/em>,<em>S<\/em>) where <em>S<\/em> is simply equal to <em>R<\/em>.  \u2610<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This defines a map \u03b7 : <em>X<\/em> \u2192 <strong>S<\/strong>(<em>X<\/em>,<strong><em>U<\/em><\/strong>).  Note that \u03b7(<em>y<\/em>) is also the set of formal balls (<em>x<\/em>,<em>R<\/em>) such that <em>R<\/em>[<em>x<\/em>] is a neighborhood of <em>y<\/em> in the induced topology of <strong><em>U<\/em><\/strong>.  Indeed, if (<em>x<\/em>,<em>R<\/em>) is in \u03b7(<em>y<\/em>), then (<em>x<\/em>,<em>R<\/em>) \u2264<sup><em><strong>U<\/strong><\/em>+<\/sup> (<em>y<\/em>,<em>S<\/em>) for some entourage <em>S<\/em>.  In particular, the neighborhood <em>S<\/em>[<em>y<\/em>] of <em>y<\/em> is included in <em>R<\/em>[<em>x<\/em>].  In the converse direction, if <em>R<\/em>[<em>x<\/em>] is a neighborhood of <em>y<\/em>, then it contains a neighborhood <em>T<\/em>[<em>y<\/em>], for some entourage <em>T<\/em>.  In particular, it contains the smaller neighborhood <em>S<\/em>[<em>y<\/em>], where <em>S<\/em> \u225d <em>R<\/em> \u2229 <em>T<\/em>.  Since <em>R<\/em> contains <em>S<\/em>, we obtain that (<em>x<\/em>,<em>R<\/em>) \u2264<sup><em><strong>U<\/strong><\/em>+<\/sup> (<em>y<\/em>,<em>S<\/em>), so that (<em>x<\/em>,<em>R<\/em>) is in \u03b7(<em>y<\/em>).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Lemma C.<\/strong>  \u03b7 is monotone and order-reflecting, that is, for all <em>x<\/em>,<em>y<\/em> in <em>X<\/em>, <em>x<\/em>\u2264<em>y<\/em> in the specialization preordering of <em>X<\/em> if and only if \u03b7(<em>x<\/em>) \u2286 \u03b7(<em>y<\/em>).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Proof.  If <em>x<\/em>\u2264<em>y<\/em>, then let us consider any element (<em>z<\/em>,<em>T<\/em>) of \u03b7(<em>x<\/em>).  Hence <em>T<\/em>[<em>z<\/em>] is a neighborhood of <em>x<\/em>, and therefore also of <em>y<\/em>, since all open subsets are upwards closed in the specialization preordering.  It follows that (<em>z<\/em>,<em>T<\/em>) is in \u03b7(<em>y<\/em>).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Conversely, let us assume that \u03b7(<em>x<\/em>) \u2286 \u03b7(<em>y<\/em>).  At the end of the proof of Lemma B, we have seen that for every entourage <em>R<\/em>, (<em>x<\/em>,<em>R<\/em>) is in \u03b7(<em>x<\/em>).  Hence (<em>x<\/em>,<em>R<\/em>) is also in \u03b7(<em>y<\/em>), meaning that <em>R<\/em>[<em>x<\/em>] is a neighborhood of <em>y<\/em>\u2014in particular, it contains <em>y<\/em>.  Since <em>R<\/em> is arbitrary, every open neighborhood of <em>x<\/em> contains <em>y<\/em>, and therefore <em>x<\/em>\u2264<em>y<\/em>.  \u2610<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In particular, if <em>X<\/em> is T<sub>0<\/sub> in the topology induced by <strong><em>U<\/em><\/strong>, then \u03b7 is injective, which is good news.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">What should we take as a quasi-uniformity on <strong>S<\/strong>(<em>X<\/em>,<strong><em>U<\/em><\/strong>)?  We do not have much choice, really: we should take the <em>largest<\/em> quasi-uniformity that makes \u03b7 uniformly continuous.  That always exists because of the following, pretty high-level statement.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Proposition.<\/strong>  The collection of all quasi-uniformities on a set <em>Z<\/em> is a complete lattice.  Given any class of maps <em>f<sub>i<\/sub><\/em> : <em>Y<sub>i<\/sub><\/em> \u2192 <em>Z<\/em>, <em>i<\/em> \u2208 <em>I<\/em>, where each <em>Y<sub>i<\/sub><\/em> is a quasi-uniform space, there is a largest quasi-uniformity on <em>Z<\/em> that makes all the maps <em>f<sub>i<\/sub><\/em> uniformly continuous.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Proof.<\/em>  For the first part, let <strong><em>U<\/em><\/strong><em><sub>j<\/sub><\/em>, <em>j<\/em> \u2208 <em>J<\/em>, be a family of quasi-uniformities on <em>Z<\/em>.  In particular, they are elements of <strong>F<\/strong>(<strong>Refl<\/strong>(<em>Z<\/em>)), the poset of filters of binary reflexive relations on <em>Z<\/em>.  For every distributive bounded lattice <em>L<\/em>, its poset <strong>F<\/strong>(<em>L<\/em>) of filters is a complete lattice, in which the supremum of a family <em>F<\/em> of filters is the collection of infima of finitely many elements from \u222a<em>F<\/em>.  In the case at hand, that means that we can build the supremum <strong><em>U<\/em><\/strong> of the filters <strong><em>U<\/em><\/strong><em><sub>j<\/sub><\/em>, <em>j<\/em> \u2208 <em>J<\/em>, inside <strong><strong>F<\/strong><\/strong>(<strong><strong>Refl<\/strong><\/strong>(<em>Z<\/em>)), as the finite intersections of binary relations <em>R<\/em><sub>1<\/sub> \u2229 &#8230; \u2229 <em>R<\/em><sub><em>n<\/em><\/sub> (<em>n<\/em>\u22650; if <em>n<\/em>=0, this is the relation relating every pair of elements, and that is indeed a reflexive relation).  It now suffices to observe that this filter <strong><em>U<\/em><\/strong> is a quasi-uniformity: then <strong><em>U<\/em><\/strong> will be the desired supremum of the family (<strong><em>U<\/em><\/strong><em><sub>j<\/sub><\/em>)<sub><em>j<\/em> \u2208 <em>J<\/em><\/sub> in the poset of quasi-uniformities on <em>Z<\/em>.  In order to show this, let <em>R<\/em> \u225d <em>R<\/em><sub>1<\/sub> \u2229 &#8230; \u2229 <em>R<\/em><sub><em>n<\/em><\/sub> be any element of <strong><em>U<\/em><\/strong>.  <em>R<\/em><sub>1<\/sub> is an element of some <strong><em>U<\/em><\/strong><em><sub>j<\/sub><\/em>, and therefore has a splitter \u00bd<em>R<\/em><sub>1<\/sub> in <strong><em>U<\/em><\/strong><em><sub>j<\/sub><\/em>.  We do the same with <em>R<\/em><sub>2<\/sub>, &#8230;, <em>R<\/em><sub><em>n<\/em><\/sub>, and we let <em>R&#8217;<\/em> be (\u00bd<em>R<\/em><sub>1<\/sub>) \u2229 &#8230; \u2229 (\u00bd<em>R<\/em><sub><em>n<\/em><\/sub>).  Since \u00bd<em>R<\/em><sub>1<\/sub>, &#8230;, \u00bd<em>R<\/em><sub><em>n<\/em><\/sub> are all in some <strong><em>U<\/em><\/strong><em><sub>j<\/sub><\/em> (not necessarily the same one), <em>R&#8217;<\/em> is in <strong><em>U<\/em><\/strong>.  Moreover, it is clear that <em>R&#8217;<\/em> o <em>R&#8217;<\/em> is included in <em>R<\/em>.  Hence <strong><em>U<\/em><\/strong> is a quasi-uniformity, as promised.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This shows that the poset of all quasi-uniformities has all suprema.  It follows that it also has all infima: in order to compute the infimum of a family <em>F<\/em>, compute the supremum of its family of lower bounds.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We turn to the second part.  Note, by the way, that we take a <em>class<\/em> of maps, not necessarily a set of maps here.  Let <em>F<\/em> be the collection of quasi-uniformities on <em>Z<\/em> that make all the maps <em>f<sub>i<\/sub><\/em> uniformly continuous.  (This is a set, even when those maps form a proper class.)  Let <strong><em>U<\/em><\/strong> be the supremum of <em>F<\/em> in the complete lattice of quasi-uniformities on <em>Z<\/em>.  We claim that <strong><em>U<\/em><\/strong> is again in <em>F<\/em>, so that <strong><em>U<\/em><\/strong> will indeed be the <em>largest<\/em> quasi-uniformity that makes all the maps <em>f<sub>i<\/sub><\/em> uniformly continuous.  To this end, let <em>R<\/em> \u225d <em>R<\/em><sub>1<\/sub> \u2229 &#8230; \u2229 <em>R<\/em><sub><em>n<\/em><\/sub> be any element of <strong><em>U<\/em><\/strong>, where each <em>R<sub>j<\/sub><\/em> is in some quasi-uniformity that occurs as an element of <em>F<\/em>.  Let us also fix an arbitrary <em>i<\/em> \u2208 <em>I<\/em>.  We must show that (<em><em>f<sub>i<\/sub><\/em><\/em>\u00d7<em><em>f<sub>i<\/sub><\/em><\/em>)<sup>\u20131<\/sup>&nbsp;(<em>R<\/em>) is an entourage of <em>Y<sub>i<\/sub><\/em>.  We realize that (<em><em>f<sub>i<\/sub><\/em><\/em>\u00d7<em><em>f<sub>i<\/sub><\/em><\/em>)<sup>\u20131<\/sup>&nbsp;(<em>R<\/em>) is equal to (<em><em>f<sub>i<\/sub><\/em><\/em>\u00d7<em><em>f<sub>i<\/sub><\/em><\/em>)<sup>\u20131<\/sup>&nbsp;(<em><em>R<\/em><\/em><sub>1<\/sub>) \u2229 &#8230; \u2229 (<em><em>f<sub>i<\/sub><\/em><\/em>\u00d7<em><em>f<sub>i<\/sub><\/em><\/em>)<sup>\u20131<\/sup>&nbsp;(<em>R<\/em><sub><em>n<\/em><\/sub>).  Now <em><em>R<\/em><\/em><sub>1<\/sub> is in some quasi-uniformity <strong><em>U<\/em><\/strong><em><sub>j<\/sub><\/em> that occurs as an element of <em>F<\/em>.  By definition of <em>F<\/em>, <em><em>f<sub>i<\/sub><\/em><\/em> is uniformly continuous from <em>Y<sub>i<\/sub><\/em> to <em>Z<\/em> with the quasi-uniformity <strong><em>U<\/em><\/strong><em><sub>j<\/sub><\/em>: so (<em><em>f<sub>i<\/sub><\/em><\/em>\u00d7<em><em>f<sub>i<\/sub><\/em><\/em>)<sup>\u20131<\/sup>&nbsp;(<em><em>R<\/em><\/em><sub>1<\/sub>) is an entourage of <em>Y<sub>i<\/sub><\/em>.  We do the same with <em>R<\/em><sub>2<\/sub>, &#8230;, <em>R<\/em><sub><em>n<\/em><\/sub>.  Therefore (<em><em>f<sub>i<\/sub><\/em><\/em>\u00d7<em><em>f<sub>i<\/sub><\/em><\/em>)<sup>\u20131<\/sup>&nbsp;(<em>R<\/em>) is a finite intersection of entourages of <em>Y<sub>i<\/sub><\/em>, and is therefore itself an entourage of <em>Y<sub>i<\/sub><\/em>.  It follows that <em>f<sub>i<\/sub><\/em> is uniformly continuous from <em>Y<sub>i<\/sub><\/em> to <em>Z<\/em> with the quasi-uniformity <strong><em>U<\/em><\/strong>.  \u2610<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The latter proposition has an interesting categorical reading.  It shows that the forgetful functor from the category <strong>QUnif<\/strong> of quasi-uniform spaces and uniformly continuous maps to <strong>Set<\/strong> is <em>topological<\/em>.  (Well, it almost does.  Let me spare the details.)  I can only recommend reading Chapter VI of [3] for an introduction to this pretty amazing piece of category theory.  It immediately follows that <strong>QUnif<\/strong> is complete, cocomplete, has regular factorizations, separators and coseparators, and (since it is also fiber-small) is also wellpowered and co-wellpowered, among other miracles (whatever all that may mean, if you do not know already).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">But let us return to our topic.  We equip <strong>S<\/strong>(<em>X<\/em>,<strong><em>U<\/em><\/strong>) with the largest quasi-uniformity <strong><em>U<\/em><\/strong><sup>+<\/sup> that makes \u03b7 : <em>X<\/em> \u2192 <strong>S<\/strong>(<em>X<\/em>,<strong><em>U<\/em><\/strong>) uniformly continuous.  I have no concrete description of that quasi-uniformity, though.  But we now have a quasi-uniform space <strong>S<\/strong>(<em>X<\/em>,<strong><em>U<\/em><\/strong>), and a uniformly continuous map \u03b7 : <em>X<\/em> \u2192 <strong>S<\/strong>(<em>X<\/em>,<strong><em>U<\/em><\/strong>), which is injective if <em>X<\/em> is T<sub>0<\/sub>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Conclusion<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Let me stop there.  This post is already long enough.  However, let me ask a few questions, which I may or may not answer in the future.<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Is there any concrete description of the quasi-uniformity <strong><em>U<\/em><\/strong><sup>+<\/sup> on <strong>S<\/strong>(<em>X<\/em>,<strong><em>U<\/em><\/strong>)?<\/li>\n\n\n\n<li>Is <strong>S<\/strong>(<em>X<\/em>,<strong><em>U<\/em><\/strong>) is isomorphic to the <em>Smyth-completion<\/em> of <em>X<\/em> [4, Section V]?<\/li>\n\n\n\n<li>By analogy with hemi-metric spaces, on may define a quasi-uniform space as complete (or &#8216;Yoneda-complete&#8217;) if and only if \u03b7 is surjective.  (I may explain why in a later post.)  Is there a more concrete way of defining that form of completeness?<\/li>\n\n\n\n<li>With that notion of completeness, is <strong>S<\/strong>(<em>X<\/em>,<strong><em>U<\/em><\/strong>) complete?  Remember that the formal ball completion of a hemi-metric case is not just (Yoneda-)complete, but even algebraic complete.<\/li>\n\n\n\n<li>Is it true that \u03b7 is an isomorphism of quasi-uniform spaces if and only if <em>X<\/em> is Smyth-complete?  That would extend similar results on hemi-metric spaces.  That would also follow from a positive solution to problem 2 in this list.<\/li>\n<\/ol>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Jean Goubault-Larrecq and Kok Min Ng.&nbsp;<a href=\"https:\/\/lmcs.episciences.org\/4100\">A few notes on formal balls<\/a>. Logical Methods in Computer Science 13(4), nov. 28, 2017.<\/li>\n\n\n\n<li>Steven Vickers.&nbsp;<a href=\"https:\/\/www.tac.mta.ca\/tac\/volumes\/14\/15\/14-15.pdf\">Localic Completion of Generalized Metric Spaces I<\/a>. Theory and Application of Categories 14(15), pages 328-356, 2005.<\/li>\n\n\n\n<li>Ji\u0159\u00ed Ad\u00e1mek, Horst Herrlich, and George E. Strecker.  <em>Abstract and concrete categories.  The joy of cats.<\/em>  John Wiley and Sons, Inc., 1990.  Online version available <a href=\"https:\/\/katmat.math.uni-bremen.de\/acc\/\">here<\/a>.<\/li>\n\n\n\n<li>Michael B. Smyth.  Quasi-uniformities: Reconciling domains with metric spaces.  In: Main M., Melton A., Mislove M., Schmidt D. (eds) Mathematical Foundations of Programming Language Semantics. MFPS 1987. Lecture Notes in Computer Science, vol 298. Springer, Berlin, Heidelberg. <a href=\"https:\/\/doi.org\/10.1007\/3-540-19020-1_12\">doi.org\/10.1007\/3-540-19020-1_12<\/a><\/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>  (January 23rd, 2021)<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Formal balls are an extraordinarily useful notion in the study of quasi-metric, and even hemi-metric spaces: see Section 7.3 of the book, and [1]. Is there any way of extending the notion to the case of quasi-uniform spaces? This is &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=3081\">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-3081","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/3081","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=3081"}],"version-history":[{"count":125,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/3081\/revisions"}],"predecessor-version":[{"id":5892,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/3081\/revisions\/5892"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3081"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}