{"id":1251,"date":"2017-09-25T11:43:39","date_gmt":"2017-09-25T09:43:39","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1251"},"modified":"2022-11-19T15:20:41","modified_gmt":"2022-11-19T14:20:41","slug":"isbells-density-theorem","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1251","title":{"rendered":"Isbell’s density theorem"},"content":{"rendered":"

When I wrote my latest blog post<\/a>, there were many things I thought would be useful to know about sublocales.\u00a0 Those eventually turned out to be useless in that context.\u00a0 However, I think they should be known, in a more general context.<\/p>\n

Remember that a sublocale is a locale-theoretic notion that best represents the notion of subspace of topological space.\u00a0 I have based several intuitive arguments on the connection between the two notions, imagining that sublocales are some kind of abstract notion of subspace.<\/p>\n

Not so: sublocales and subspaces can be very<\/em> different.\u00a0 The latest post is an illustration of that, and today we shall see two other cases where caution has to be exerted, and which are perhaps simpler:<\/p>\n

    \n
  1. Isbell’s density theorem;<\/li>\n
  2. The difference between intersections of sublocales and intersections of subspaces.<\/li>\n<\/ol>\n

    Isbell’s density theorem<\/h2>\n

    Given a frame \u03a9, we can form the residuation (or intuitionistic implication operator) \u21d2: u<\/em> \u21d2 v<\/em> is the largest element w<\/em> such that u<\/em> \u22c0 w<\/em> \u2264 v<\/em>.<\/p>\n

    When v<\/em> = \u22a5, we obtain intuitionistic negation<\/em> \u00acu<\/em>. When \u03a9 is the frame of opens of a topological space, \u00acU<\/em> is the interior of the complement of U<\/em> \u2014 not the complement of U<\/em>, which would in general fail to be open.<\/p>\n

    Let S<\/em>0<\/sub> be the family {\u00acu<\/em> | u<\/em> \u2208 \u03a9}. This is a sublocale. Let us check this. S<\/em>0<\/sub> is closed under arbitrary infima because the intuitionistic negation of \u22c1i \u2208 I<\/sub><\/em> ui \u2208 I<\/sub><\/em> is \u22c0i \u2208 I<\/sub><\/em> \u00acui \u2208 I<\/sub><\/em> (beware: the dual property is wrong: the negation of an infimum need not be the corresponding sup of negations). And for every element \u00acu<\/em> of S<\/em>0<\/sub>, for every v<\/em> in \u03a9 v<\/em> \u21d2 \u00acu<\/em> is equal to \u00ac(v<\/em> \u22c0 u<\/em>), hence is in S<\/em>0<\/sub>.<\/p>\n

    S<\/em>0<\/sub> is a very particular sublocale, as we shall see. By the way, Picado and Pultr [2] call it BL<\/sub>.<\/em><\/p>\n

    Every sublocale S<\/em> has a closure<\/em> cl(S<\/em>), which is simply equal to the smallest closed sublocale c<\/strong>(u<\/em>) containing S<\/em>. Recall that c<\/strong> maps suprema to infima, so this is equal to the intersection of all the closed sublocales c<\/strong>(u<\/em>) that contain S<\/em>, or equivalently this is c<\/strong>(u<\/em>) where u<\/em> is the supremum of all the elements v<\/em> of \u03a9 such that S<\/em> \u2286 c<\/strong>(v<\/em>).<\/p>\n

    Recall also that c<\/strong>(u<\/em>) is simply the upward closure \u2191u<\/em>. For example, c<\/strong>(\u22a5), the largest closed sublocale, is just \u03a9 itself, the largest sublocale at all.<\/p>\n

    Call a sublocale S<\/em> dense<\/em> if and only if cl(S<\/em>) is the largest possible sublocale, \u03a9 itself. In other words, S<\/em> is dense if and only if the only u<\/em> in \u03a9 such that S<\/em> \u2286 \u2191u<\/em> is \u22a5.<\/p>\n

    We can simplify the definition further. For every sublocale S<\/em>, the infimum of all the elements of S<\/em> is again in S<\/em>, since S<\/em> is closed under arbitrary infima. This shows that every sublocale has a least element u<\/em>. Clearly S<\/em> \u2286 \u2191u<\/em>, so if S<\/em> is dense, then u<\/em>=\u22a5. Conversely, any sublocale containing \u22a5 is dense.<\/p>\n

    It follows that:<\/p>\n

    A sublocale S<\/em> is dense if and only if it contains \u22a5.<\/p><\/blockquote>\n

    Immediately, we obtain that S<\/em>0<\/sub> is a dense sublocale: indeed \u22a5 is the intuitionistic negation of \u22a4, \u00ac\u22a4.<\/p>\n

    More strangely \u2014 this is Isbell’s density theorem (see III.8.3 in [2]):<\/p>\n

    Lemma.<\/strong> S<\/em>0<\/sub> is the smallest dense sublocale.
    \nProof<\/em>. Imagine S<\/em> is another dense sublocale. S<\/em> contains \u22a5, and since S<\/em> contains v<\/em> \u21d2 u<\/em> for every v<\/em> in \u03a9 and every u<\/em> in S<\/em>, then by taking u<\/em>=\u22a5, it contains \u00acv<\/em> for every v<\/em> in \u03a9. In other words, S<\/em>0<\/sub> is included in S<\/em>. \u2610<\/p>\n

    That has absolutely no equivalent in the realm of topological spaces: if there is a smallest dense subset, then it is the intersection of all dense subsets of a space; but that is in general not dense. For example, if you start from a space X<\/em> where no point is isolated (i.e., the space is dense-in-itself<\/em>; a point x<\/em> is isolated if {x<\/em>} is open), then X<\/em>-{x<\/em>} is dense for every point x<\/em>, and the intersection of those sets when x<\/em> varies is empty\u2014certainly not dense, unless X<\/em> itself is empty. The space R<\/strong> of real numbers with its usual T2<\/sub> topology, for one, is dense-in-itself, and has no smallest dense subset.<\/p>\n

    In the pointfree version of R<\/strong>, however, the smallest dense sublocale S<\/em>0<\/sub> is the collection of interiors of closed sets of R<\/strong>, or equivalently, the collection of regular open subsets of R (see Exercises 8.1.7 and 8.1.8 in the book<\/a>). There are plenty of them! Every open interval of R is regular open, for example. In particular S<\/em>0<\/sub> is very different from the smallest locale, which is just {\u22a4} (where \u22a4=X<\/em> itself).<\/p>\n

    Intersection of sublocales and sublocales of intersections<\/h2>\n

    We apply that to elucidate the relationship between intersection of sublocales and intersection of subspaces.<\/p>\n

    Let us recall that the set of sublocales<\/a> Sl<\/strong>(\u03a9) of a frame \u03a9 forms a coframe under inclusion.\u00a0 The greatest lower bound of two sublocales S<\/em>1<\/sub> and S<\/em>2<\/sub> is simply their intersection.<\/p>\n

    In particular, the intersection of two sublocales S<\/em>1<\/sub> and S<\/em>2<\/sub> is another sublocale. One may think of that construction as the analogue of taking the intersection of two subspaces of a topological space, but one should really be cautious here; in fact the analogy breaks pretty quickly, as we shall see.<\/p>\n

    Any subspace A<\/em> of a topological space X<\/em> gives rise to a sublocale of O<\/strong>(X<\/em>), given by SA<\/sub><\/em> = {U<\/em> \u2208 O<\/strong>(X<\/em>) | U<\/em> is the largest open subset of X<\/em> whose intersection with A<\/em> equals U<\/em> \u2229 A<\/em>}. In other words, SA<\/sub><\/em> is the set of fixed points Fix(\u03bdA<\/sub><\/em>) of the nucleus \u03bdA<\/sub><\/em> mapping each U<\/em> \u2208 O<\/strong>(X<\/em>) to the largest open subset V<\/em> of X<\/em> such that V<\/em> \u2229 A<\/em>=U<\/em> \u2229 A<\/em>.<\/p>\n

    SA<\/sub><\/em> is, naturally, order-isomorphic to O<\/strong>(A<\/em>). We have:<\/p>\n

    Lemma.<\/strong> The mapping A<\/em> \u21a6 SA<\/sub><\/em> is monotonic.
    \nProof<\/em>. Assume A<\/em> \u2286 B<\/em>. We must show that SA<\/sub><\/em> \u2286 SB<\/sub><\/em>. Since the coframe of sublocales is isomorphic to the opposite of the frame of nuclei, it is equivalent to show that \u03bdB<\/sub><\/em> \u2264 \u03bdA<\/sub><\/em>. For every open U<\/em>, \u03bdB<\/sub><\/em> (U<\/em>) is the largest open set V<\/em> such that V<\/em> \u2229 B<\/em>=U<\/em> \u2229 B<\/em>. Taking intersections with A<\/em>, and recalling that B<\/em> \u2229 A<\/em> = A<\/em> since A<\/em> \u2286 B<\/em>, we obtain V<\/em> \u2229 A<\/em>=U<\/em> \u2229 A<\/em>. Hence V<\/em> = \u03bdB<\/sub><\/em> (U<\/em>) is an open subset whose intersection with A<\/em> equals U<\/em> \u2229 A<\/em>, and is therefore certainly included in the largest such open subset, \u03bdA<\/sub><\/em> (U<\/em>). \u2610<\/p>\n

    Given two subspaces A<\/em> and B<\/em> of X<\/em>, how do SA<\/sub><\/em> \u2229 SB<\/sub><\/em> and SA \u2229 B<\/sub><\/em> compare? The following is easy.
    \n Lemma.<\/strong> SA <\/sub><\/em>\u2229<\/sub> B<\/sub><\/em> is included in SA<\/sub><\/em> \u2229 SB<\/sub><\/em>.
    \nProof<\/em>. It suffices to show that SA \u2229 B<\/sub><\/em> is included both in SA<\/sub><\/em> and in SB<\/sub><\/em>, which follows from the previous Lemma. \u2610<\/p>\n

    In general, SA \u2229 B<\/sub><\/em> and SA<\/sub><\/em> \u2229 SB<\/sub><\/em> differ, and can in fact differ by a very large margin. Consider the subspace A<\/em>=Q<\/strong> of R<\/strong> consisting of the rational numbers, and the complementary subspace B<\/em> of irrational numbers. Clearly A<\/em> \u2229 B<\/em> is empty, so SA \u2229 B<\/sub><\/em> is the smallest sublocale {\u22a4}. But A<\/em> and B<\/em> are both dense (in the usual sense) in R<\/strong>, so the sublocales SA<\/sub><\/em> and SB<\/sub><\/em> are both dense, in the sense we have just given. By Isbell’s density theorem, they both contain S<\/em>0<\/sub>, and we have seen that it is much larger than {\u22a4}.<\/p>\n

    How should one view locales and sublocales?<\/h2>\n

    A normal first reaction to that kind of result would be: `this cannot be the right theory, let us forget about it’.\u00a0 I am sometimes tempted to abide by this.\u00a0 However, the mathematics of locales is so beautiful that there should be some truth in it.<\/p>\n

    Here is one possible explanation of the difference between sublocales and subspaces.\u00a0 Instead of considering points as the primary ingredients of a space, and opens as some kind of structure that would play a useful, but secondary role, let us recognize that spaces contain both points and some glue<\/em> between the points.\u00a0 (I don’t remember where I have seen this analogy.\u00a0 The idea is certainly not mine, perhaps Johnstone’s or Isbell’s.)\u00a0 The opens, and notions of continuity are here to express how elastic that glue is.<\/p>\n

    Topology is one extreme, where spaces are spaces of points, and, oh, in passing, they are glued together in some way.<\/p>\n

    Locale theory is another extreme, where you recognize a space as a big amount of glue, and, oh, in passing, that glue sometimes connects infinitesimal things called points.<\/p>\n

    A point of view that reconciles the two is the equivalence between sober spaces and spatial lattices: points exist, and so does glue (the topology).\u00a0 Moreover, glue gives structure to points, but also, conversely, points give structure to the glue, in the case of spatial lattices.\u00a0 Chu spaces<\/a> are one way of representing both on an equal footing, and have been actively promoted by Vaughn Pratt since 1994.<\/p>\n

      \n
    1. John Isbell. Product spaces in locales<\/a>. Proceedings of the American Mathematical Society, 81(1), January 1981.<\/li>\n
    2. Jorge Picado and Ale\u0161 Pultr. Frames and locales \u2014 topology without points. Birkh\u00e4user, 2010.<\/li>\n<\/ol>\n

      \u2014 Jean Goubault-Larrecq<\/a>\u00a0(September 25th, 2017)\"jgl-2011\"<\/p>\n","protected":false},"excerpt":{"rendered":"

      When I wrote my latest blog post, there were many things I thought would be useful to know about sublocales.\u00a0 Those eventually turned out to be useless in that context.\u00a0 However, I think they should be known, in a more … Continue reading →<\/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-1251","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1251","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=1251"}],"version-history":[{"count":7,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1251\/revisions"}],"predecessor-version":[{"id":5935,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1251\/revisions\/5935"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1251"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}