{"id":1534,"date":"2018-08-22T17:22:46","date_gmt":"2018-08-22T15:22:46","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1534"},"modified":"2022-11-19T15:15:51","modified_gmt":"2022-11-19T14:15:51","slug":"projective-limits-of-topological-spaces-i-oddities","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1534","title":{"rendered":"Projective limits of topological spaces I: oddities"},"content":{"rendered":"

I like to explain projective limits as follows. \u00a0Imagine you take a photo of some landscape with an old, low-resolution camera. \u00a0You can vaguely recognize the landscape, but the image is somehow blurred. \u00a0Hence you decide to use a second camera, with better resolution: each pixel in the first picture now corresponds, say, to a square of four pixels in the second picture. \u00a0The image is better, but not perfect, so you decide to use an even better camera, and so on. \u00a0Intuitively, if you were able to build a limit of that series of pictures, you would obtain a perfect image of the landscape in the end. \u00a0That is the projective limit of the sequence of pictures you have taken.<\/p>\n

The definition of projective limits<\/h2>\n

Formally, we consider a family of spaces Xi<\/sub><\/em> indexed by some i<\/em> \u2208 I<\/em> (those are our pictures), we assume that\u00a0I<\/em> is preordered and directed, and that, for all\u00a0i<\/em>\u2264j<\/em> in\u00a0I<\/em>, there is a so-called\u00a0bonding map<\/em> pij<\/sub><\/em> from Xj<\/sub><\/em> to Xi<\/sub><\/em>: for every point\u00a0x<\/em> in Xj<\/sub><\/em>, pij<\/sub><\/em> (x<\/em>) gives you the position of pixel where\u00a0x<\/em> would lie in the coarser picture Xi<\/sub><\/em>. \u00a0We require that pii<\/sub><\/em> \u00a0be the identity map for every\u00a0i<\/em>, and that pij<\/sub><\/em> o pjk<\/sub><\/em> = pik<\/sub><\/em>\u00a0for all i<\/em>\u2264j<\/em>\u2264k<\/em>. \u00a0The data of all spaces Xi<\/sub><\/em> and all bonding maps pij<\/sub><\/em>\u00a0is called a\u00a0projective system<\/em> of spaces.<\/p>\n

Categorically, a projective system is just a functor from\u00a0I<\/em>, seen as a (thin) category, to the category\u00a0Top<\/strong> of topological spaces. \u00a0This construction works with any category instead of\u00a0Top<\/strong>, and we can define projective systems of sets, of groups, etc. \u00a0We might even replace\u00a0I<\/em> with an arbitrary category, but the case where I<\/em> is\u00a0a directed preordered set is important in mathematics, and there are some results which require that restriction (notably Prohorov’s construction of a projective limit of measures<\/em>).<\/p>\n

The\u00a0projective limit<\/em> of a projective system is just a limit of that functor. \u00a0Explicitly, a\u00a0cone<\/em> over the projective systems\u00a0is a space\u00a0X<\/em>, together with (“projection”) maps\u00a0pi<\/sub><\/em>\u00a0:\u00a0X<\/em>\u00a0\u2192 Xi\u00a0<\/sub><\/em>for each\u00a0i<\/em>, such that\u00a0pij<\/sub><\/em> o pj<\/sub><\/em> = pi<\/sub><\/em>\u00a0for all i<\/em>\u2264j<\/em>. And a projective limit is a universal cone, namely a cone (X<\/em>, pi<\/sub><\/em>) such that for every cone\u00a0(Y<\/em>, qi<\/sub><\/em>), there is a unique morphism\u00a0f<\/em> from\u00a0Y<\/em> to\u00a0X<\/em> such that pi<\/sub><\/em> o f<\/i>\u00a0= qi<\/sub><\/em>\u00a0for every\u00a0i<\/em>. \u00a0That is the usual category-theoretic definition, but if you are not that categorically minded, that will not tell you much.<\/p>\n

Instead, let me describe one particular projective limit, the canonical projective limit X<\/em>\u00a0of the projective system (Xi<\/sub><\/em>,\u00a0pij<\/sub><\/em>) (in\u00a0Top<\/strong>). \u00a0Its elements are the tuples (xi<\/sub><\/em>)i\u2208 I<\/sub><\/em> where each xi<\/sub><\/em> is in Xi<\/sub><\/em>, and where\u00a0xi<\/sub><\/em> = pij<\/sub><\/em> (xj<\/sub><\/em>) for all\u00a0i<\/em>\u2264j<\/em> in\u00a0I<\/em>. \u00a0This makes\u00a0X<\/em> a subset of the product space\u00a0\u03a0i\u2208 I<\/sub><\/em> Xi<\/sub><\/em>, and we topologize it with the subspace topology. \u00a0The projection map pi<\/sub><\/em>\u00a0:\u00a0X<\/em>\u00a0\u2192 Xi\u00a0<\/sub><\/em>maps every such tuple to xi<\/sub><\/em>.<\/p>\n

For those accustomed to domain theory…<\/h2>\n

There is a similar construction in domain theory, where in addition to bonding maps pij<\/sub><\/em>,we have maps eij<\/sub><\/em>\u00a0running in the other direction, from\u00a0Xi<\/sub><\/em>, to\u00a0Xj<\/sub><\/em>, and such that pij\u00a0<\/sub><\/em>o eij\u00a0<\/sub><\/em>= id, eij\u00a0<\/sub><\/em>o pij\u00a0<\/sub><\/em>\u2264 id (for all\u00a0i<\/em>\u2264j<\/em>), \u00a0eii<\/sub><\/em>\u00a0= id\u00a0and\u00a0ejk<\/sub><\/em> o eij<\/sub><\/em> = eik<\/sub><\/em>\u00a0for all i<\/em>\u2264j<\/em>\u2264k<\/em>. \u00a0That is called an\u00a0ep-system<\/em> in the book<\/a> (Section 9.6).<\/p>\n

The projective limit of an ep-system is the projective limit X<\/em>\u00a0of the underlying projective system, forgetting about the maps eij<\/sub><\/em>. \u00a0That construction has plenty of nice properties. \u00a0Here is a list of interesting properties we have in that case\u2014I will leave the task of proving them to you:<\/p>\n

    \n
  1. For every\u00a0i<\/em> in\u00a0I<\/em>, there is a map\u00a0ei<\/sub><\/em>\u00a0:\u00a0Xi<\/sub><\/em>\u00a0\u2192 X<\/em>, defined as follows. \u00a0The\u00a0j<\/em>th component of ei<\/sub><\/em>\u00a0(x<\/em>) is pjk<\/sub><\/em>(eik<\/sub><\/em>(x<\/em>)), where\u00a0k<\/em> is any index above both\u00a0i<\/em> and\u00a0j<\/em>. (That k<\/em> exists because\u00a0I<\/em> is directed. \u00a0I will let you\u00a0check that this is independent of\u00a0k.<\/em>) \u00a0It is continuous, and pi\u00a0<\/sub><\/em>o ei\u00a0<\/sub><\/em>= id, ei\u00a0<\/sub><\/em>o pi\u00a0<\/sub><\/em>\u2264 id.<\/li>\n
  2. For all\u00a0i<\/em>\u2264j<\/em> in\u00a0I<\/em>, ej\u00a0<\/sub><\/em>o eij<\/sub><\/em>\u00a0= ei<\/sub><\/em>.<\/li>\n
  3. For every point\u00a0x<\/em> in the projective limit\u00a0X<\/em>, the family of points ei<\/sub><\/em>(pi<\/sub><\/em>(x<\/em>)) forms a monotone net (hence is directed), and its supremum is\u00a0x<\/em>.<\/li>\n
  4. For every open subset\u00a0U<\/em> of\u00a0X<\/em>, the family of open subsets (ei<\/sub><\/em> o pi<\/sub><\/em>)-1<\/sup> (U<\/em>) is also a monotone net, and its union is exactly\u00a0U<\/em>.<\/li>\n<\/ol>\n

    Note that, according to intuition,\u00a0the projective limit is big: through the injective maps ei<\/sub><\/em>, it contains every Xi<\/sub><\/em>. \u00a0In particular, as soon as some Xi<\/sub><\/em>\u00a0is non-empty, then\u00a0X<\/em> is non-empty.<\/p>\n

    Also, the projection maps pi<\/sub><\/em>\u00a0are all surjective. \u00a0Indeed, they are retractions.<\/p>\n

    None of that will survive for general projective limits, as we will now see.<\/p>\n

    A first oddity<\/h2>\n

    We assume that every Xi<\/sub><\/em>\u00a0is non-empty. \u00a0Is the projective limit necessarily non-empty? \u00a0The domain-theoretic intuition for the last section should tell us it must be so, but that is not true. \u00a0Here is a simple example.<\/p>\n

    Example 1.<\/strong> We take\u00a0I<\/em>=N<\/strong> with its usual ordering,\u00a0Xn<\/sub><\/em>\u00a0= the open interval (0, 1) in\u00a0R<\/strong>, and for every pair\u00a0m<\/em>\u2264n<\/em>,\u00a0pmn<\/sub><\/em>\u00a0(x<\/em>) = x<\/em>\/2n\u2014m<\/sup><\/em>. \u00a0It is easy to see from the explicit construction of the projective limit as a space of tuples that the projective limit is empty: there is no tuple of numbers in (0, 1) such that each one is twice as large as the previous one in the list.<\/p>\n

    All right, but we cheated: in Example 1, the bonding maps are not surjective. \u00a0Hence, by going from Xn<\/sub><\/em>\u00a0to Xn+<\/sub><\/em>1<\/sub>, we lose a fraction of (1\u20142n\u2014m<\/sup><\/em>) of all the points. \u00a0In the limit, we have lost of all them: that should have been expected.<\/p>\n

    Let us refine our question: if the spaces Xi<\/sub><\/em>\u00a0are non-empty\u00a0and<\/em> all the bonding maps pij<\/sub><\/em>\u00a0are surjective, is the projective limit non-empty?<\/p>\n

    Leon Henkin was the first to answer the question [1], in the negative. \u00a0His paper is pretty complex, and involves a sophisticated construction based on ordinals.<\/p>\n

    Waterhouse later realized that there was a much simpler solution [2]. \u00a0Indeed, his paper is 6\u00a0lines<\/em> long! \u00a0(excluding title, author, and bibliography; I will explain it in slightly more detail). \u00a0Here it is.<\/p>\n

    Example 2.<\/strong>\u00a0We consider a projective limit of sets, not topological spaces here, because topology is in fact irrelevant in that case. \u00a0If needed, put the discrete topology on all the sets involved in the argument.
    \nLet A<\/em> be an uncountable set, such as R<\/strong> for example. Let\u00a0I<\/em> be the lattice of finite subsets of\u00a0A<\/em>, ordered by inclusion. For each i<\/em> \u2208 I<\/em>, i<\/em> is a finite subset of A<\/em>, and we define Xi<\/sub><\/em>\u00a0as the set of all injective maps from\u00a0i<\/em> to\u00a0N<\/strong>. \u00a0For all\u00a0i<\/em>\u2264j<\/em> (i.e.,\u00a0i<\/em>\u2286j) <\/em>in\u00a0I<\/em>, we define\u00a0pij<\/sub><\/em>\u00a0as mapping every\u00a0f<\/em> in Xj<\/sub><\/em>\u00a0(an injective map from j<\/em> to\u00a0N<\/strong>)\u00a0to its restriction f|i<\/sub><\/em>\u00a0to the subset i<\/em>.\u00a0 This defines a projective system of sets with surjective bonding maps.
    \nGiven any element (fi<\/sub><\/em>)i\u2208 I<\/sub><\/em> of the canonical projective limit X of that system, we can “glue together” the maps fi<\/sub><\/em>\u00a0to obtain a single map\u00a0f<\/em> from\u00a0A<\/em> to\u00a0N<\/strong>: for every\u00a0a<\/em> in\u00a0A<\/em>, all the maps\u00a0fi<\/sub><\/em>\u00a0such that\u00a0a<\/em> belongs to\u00a0i<\/em> must map it to the same element, and this is\u00a0f<\/em>(a<\/em>) by definition. \u00a0Then\u00a0f<\/em> is injective: for all\u00a0a<\/em>\u2260b<\/em> in\u00a0A<\/em>,\u00a0f<\/em>(a<\/em>) = f<\/i>{a<\/i>,b<\/i>}<\/sub>(a<\/em>) \u2260\u00a0f<\/i>{a<\/i>,b<\/i>}<\/sub>(b<\/em>) =\u00a0f<\/em>(b<\/em>), since f<\/i>{a<\/i>,b<\/i>}<\/sub>\u00a0is injective. \u00a0However, there is no injective map from\u00a0A<\/em> to\u00a0N<\/strong>, since\u00a0A<\/em> is uncountable. \u00a0It follows that X is empty.<\/p>\n

    The case of cofinal countable chains<\/h2>\n

    It is known that such an oddity does not happen if\u00a0I<\/em> is countable (which the\u00a0I<\/em> of Example 2 is not), or more generally when I<\/em> has a cofinal monotone sequence i<\/em>0<\/sub> \u2264 i<\/em>1<\/sub> \u2264 … \u2264 ik<\/sub><\/em> \u2264 … (cofinal means that every element of\u00a0I<\/em> is below some ik<\/sub><\/em>); the latter is the case with I<\/em>=R<\/strong> with its usual ordering, for example. \u00a0I will just give a sketch of a proof.<\/p>\n

    An easy check shows that the projective limit of the projective system\u00a0(Xi<\/sub><\/em>,\u00a0pij<\/sub><\/em>) (with\u00a0i<\/em>\u2264j<\/em> in I<\/em>) is isomorphic to the projective limit of the subsystem\u00a0(Xi<\/sub><\/em>,\u00a0pij<\/sub><\/em>) with\u00a0i<\/em>\u2264j<\/em> in the chosen cofinal monotone sequence. \u00a0Replacing I<\/em> with\u00a0that monotone sequence if necessary, and reindexing, we can therefore assume that\u00a0I<\/em>\u00a0is N<\/strong> itself. \u00a0Now pick a point\u00a0x<\/em>0<\/sub>\u00a0from X<\/em>0<\/sub>, then\u00a0a point\u00a0x<\/em>1<\/sub>\u00a0from X<\/em>1<\/sub>\u00a0such that\u00a0p<\/em>01<\/sub>(x<\/em>1<\/sub>)=x<\/em>0<\/sub> (remember that p<\/em>01<\/sub>\u00a0is surjective), then\u00a0a point\u00a0x<\/em>2<\/sub>\u00a0from X<\/em>2<\/sub>\u00a0such that\u00a0p<\/em>12<\/sub>(x<\/em>2<\/sub>)=x<\/em>1<\/sub> (p<\/em>12<\/sub>\u00a0is surjective), etc. \u00a0The tuple of points xn<\/sub><\/em>\u00a0thus obtained is an element of the projective limit.<\/p>\n

    One may “generalize” that result to the case where\u00a0I<\/em> has a cofinal countable set A<\/em>\u00a0(not necessarily totally ordered). \u00a0But in that case,\u00a0I<\/em> must in fact have a cofinal monotone sequence, so that is no genuine generalization: enumerate A<\/em> as a<\/em>0<\/sub>,\u00a0a<\/em>1<\/sub>,\u00a0a<\/em>2<\/sub>, … then define\u00a0i<\/em>0<\/sub>\u00a0as a<\/em>0<\/sub>, i<\/em>1<\/sub>\u00a0as some element of I<\/em> above\u00a0i<\/em>0<\/sub>\u00a0and a<\/em>1<\/sub>\u00a0(using directedness),\u00a0i<\/em>2<\/sub>\u00a0as some element of I<\/em> above\u00a0i<\/em>1<\/sub>\u00a0and a<\/em>2<\/sub>\u00a0(using directedness), \u00a0etc.<\/p>\n

    Further oddities<\/h2>\n

    Non-emptiness is one thing, and compactness is another. \u00a0We may hope that a projective limit of compact spaces is compact\u00a0(as usual, we understand compactness without Hausdorffness). \u00a0Again, that is\u00a0not<\/em> true.<\/p>\n

    To be precise, it is true that a projective limit of compact\u00a0Hausdorff<\/em> spaces is compact (and Hausdorff). \u00a0That had been known since Bourbaki, and we will see a generalization of that result next time (from [4]).<\/p>\n

    In the meantime, here is a counterexample to the claim that a projective limit of compact spaces is compact. \u00a0This is due to A. H. Stone [3, Example 3], and it even shows that even a projective limit of Noetherian T1\u00a0<\/sub>spaces (remember that Noetherianness\u2014Section 9.7 of the book<\/a>\u2014is a very strong compactness property), with bonding maps that are even bijective, can fail to be compact.<\/p>\n

    Example 3.<\/strong> \u00a0For every n<\/em> \u2208 N<\/strong>, let Xn<\/sub><\/em>\u00a0be N<\/strong> with the following topology, akin to the cofinite topology: its closed subsets C<\/em> are those subsets such that C<\/em> \u2229 \u2191n<\/em>\u00a0is finite or equal to the whole of \u2191n<\/em>. \u00a0For all m<\/em>\u2264n\u00a0<\/em>in N<\/strong>, we simply define pmn\u00a0<\/sub><\/em>: Xn<\/sub><\/em> \u2192Xm<\/sub><\/em>\u00a0as the identity map. Every Xn<\/sub><\/em> is compact, in fact Noetherian: every filtered family of closed sets is stationary (i.e., has a least element), and is also T1<\/sub>\u00a0(the closure of every point is the point itself). Every pmn<\/sub><\/em> is surjective, in fact even bijective. I will leave to you the task of checking that pmn<\/sub><\/em> is continuous, and that the canonical projective limit of the projective system we have just defined is isomorphic to N<\/strong> with the discrete<\/em> topology, and that is not compact.<\/p>\n

    Next time<\/h2>\n

    … we will see that projective limits of compact\u00a0sober<\/em> spaces are compact (and sober). \u00a0I found this in a pretty big, recent paper on so-called rigid geometry by Fujiwara and Kato [4, Theorem 2.2.20], who call it Steenrod’s theorem.<\/p>\n

    Steenrod indeed had a similar theorem [5, Theorem 2.1], but instead claims that projective limits of compact T1<\/sub>\u00a0(instead of sober) spaces are compact. \u00a0The latter claim is faulty, as Example 3 above shows. \u00a0The error seems to lie in the very first line of the proof of Steenrod’s Theorem 2.1, which ascertains that a certain set is closed, a fact that would be true if the bonding maps were closed maps, not just continuous maps. \u00a0(The case of closed continuous bonding maps is exactly the subject of A. H. Stone’s paper [3].) \u00a0It does not seem that this has any impact on the rest of Steenrod’s results, whose spaces will anyway be Hausdorff, from what I can see.<\/p>\n

    The proof that projective limits of compact sober spaces are compact, and which, according to Fujiwara and Kato, is due to O. Gabber, is pretty clever, but it would take too much time to explain it now. \u00a0Next time, then!<\/p>\n

      \n
    1. Henkin, Leon. 1950. A Problem on Inverse Mapping Systems. Proceedings of the American Mathematical Society, 1, 224\u2013225.<\/li>\n
    2. Waterhouse, William C. 1972. An Empty Inverse Limit. Proceedings of the American Mathematical Society, 36(2), 618.<\/li>\n
    3. Stone, Arthur Harold. 1979. Inverse Limits of Compact Spaces. General Topology and its Applications, 10, 203\u2013211.<\/li>\n
    4. Fujiwara, Kazuhiro, and Kato, Fumiharu. 2017 (Feb.). Foundations of Rigid Geometry I. arXiv 1308.4734<\/a>, v5.<\/li>\n
    5. Steenrod, Norman E. 1936. Universal Homology Groups. American Journal of Mathematics, 58(4), 661\u2013701.<\/li>\n<\/ol>\n

      \u2014 Jean Goubault-Larrecq<\/a>\u00a0(August 22nd, 2018)\"jgl-2011\"<\/p>\n","protected":false},"excerpt":{"rendered":"

      I like to explain projective limits as follows. \u00a0Imagine you take a photo of some landscape with an old, low-resolution camera. \u00a0You can vaguely recognize the landscape, but the image is somehow blurred. \u00a0Hence you decide to use a second … 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-1534","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1534","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=1534"}],"version-history":[{"count":13,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1534\/revisions"}],"predecessor-version":[{"id":5926,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1534\/revisions\/5926"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1534"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}