{"id":1548,"date":"2018-09-23T16:07:37","date_gmt":"2018-09-23T14:07:37","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1548"},"modified":"2022-11-19T15:15:41","modified_gmt":"2022-11-19T14:15:41","slug":"projective-limits-of-topological-space-ii-steenrods-theorem","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1548","title":{"rendered":"Projective limits of topological space II: Steenrod’s theorem"},"content":{"rendered":"

Last time<\/a>, I explained some of the strange things that happen with projective limits of topological spaces: they can be empty, even if all the spaces in the given projective system are non-empty and all bonding maps are surjective, and they can fail to be compact, even if all the spaces in the projective system are compact.<\/p>\n

Steenrod [1, Theorem 2.1] claimed that if all the spaces in the projective system are compact and T1<\/sub>, then the projective limit is non-empty and compact T1<\/sub>. \u00a0(The\u00a0T1<\/sub>\u00a0condition is not mentioned explicitly, because what Steenrod includes the T1<\/sub>\u00a0separation axiom in his definition of topological spaces. \u00a0Also, note that he calls bicompact what we call compact.)<\/p>\n

As I said last time, this is wrong. \u00a0But Fujiwara and Kato prove [3, Theorem 2.2.20] the following\u2014and call it Steenrod’s theorem:<\/p>\n

Theorem.<\/strong> The projective limit of a projective system (pij\u00a0<\/sub><\/em>: Xj<\/sub><\/em>\u00a0\u2192 Xi<\/sub><\/em>)i\u2264j \u2208 I<\/sub><\/em> of compact sober spaces is compact and sober. It is non-empty if every Xi<\/sub><\/em> is non-empty.<\/p>\n

Note that we do not need the bonding maps pij<\/sub><\/em>\u00a0to be surjective for that to hold.<\/p>\n

My goal is to explain the proof (credited to O. Gabber by Fujiwara and Kato). \u00a0We will do it in (at least) two passes. \u00a0This month, we will prove the following apparently very special case. \u00a0We will explain the rest next month<\/p>\n

Proposition.<\/strong> \u00a0Let\u00a0 (pij\u00a0<\/sub><\/em>: Xj<\/sub><\/em>\u00a0\u2192 Xi<\/sub><\/em>)i\u2264j \u2208 I<\/sub><\/em>\u00a0be a projective system of compact sober spaces. \u00a0If every\u00a0Xi<\/sub><\/em>\u00a0is non-empty, then its projective limit\u00a0X<\/em> is non-empty as well.<\/p>\n

Recall that\u00a0X<\/em> consists of\u00a0the 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>. \u00a0X<\/em>\u00a0is topologized as a subspace of the product space\u00a0\u03a0i\u2208 I<\/sub><\/em> Xi<\/sub><\/em>. \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

The proof is pretty subtle, and will occupy the rest of this post. \u00a0Because of sobriety, instead of picking one element\u00a0xi<\/sub><\/em>\u00a0from each Xi<\/sub><\/em>, it is enough to find an irreducible closed subset\u00a0Zi<\/sub><\/em>\u00a0in each Xi<\/sub><\/em>\u2014subject to some conditions\u2014instead. \u00a0That Zi<\/sub><\/em>\u00a0will be the closure of a unique point by sobriety, and that will be xi<\/sub><\/em>.<\/p>\n

The dcpo\u00a0\u03a6<\/h2>\n

Let \u03a6 be the collection of all families\u00a0(Ci<\/sub><\/em>)i\u2208 I<\/sub><\/em> where each Ci<\/sub><\/em>\u00a0is closed and non-empty in Xi<\/sub><\/em>. \u00a0and such that the image of Cj<\/sub><\/em>\u00a0by pij\u00a0<\/sub><\/em>is included in Ci<\/sub><\/em>\u00a0for all\u00a0i<\/em>\u2264j<\/em> in\u00a0I<\/em>. \u00a0We order\u00a0\u03a6 by pointwise reverse inclusion, namely (Ci<\/sub><\/em>)i\u2208 I<\/sub><\/em>\u2264(C’i<\/sub><\/em>)i\u2208 I<\/sub><\/em> if and only if Ci<\/sub><\/em>\u00a0contains C’i<\/sub><\/em>\u00a0for every i<\/em> in\u00a0I<\/em>. \u00a0The idea is to find a maximal element of\u00a0\u03a6, and to show that it fits.<\/p>\n

In order to show that\u00a0\u03a6 has a maximal element, we plan to use Zorn’s Lemma. \u00a0We first check that\u00a0\u03a6 is non-empty: (Xi<\/sub><\/em>)i\u2208 I<\/sub><\/em>\u00a0is an element of \u03a6.<\/p>\n

By Zorn’s Lemma,\u00a0\u03a6 has a maximal element. \u00a0Let us call it\u00a0(Zi<\/sub><\/em>)i\u2208 I<\/sub><\/em>\u00a0in the sequel.<\/p>\n

The key lemma<\/h2>\n

Given any property\u00a0P<\/em> of indices\u00a0i<\/em> in\u00a0I<\/em>, we will say “for\u00a0cofinally many\u00a0j<\/em>,\u00a0P<\/em>(j<\/em>)” or “P<\/em>(j<\/em>) holds for cofinally many\u00a0j<\/em>” to say that for every\u00a0k<\/em> in\u00a0I<\/em>, there is a\u00a0j<\/em>\u2265k<\/em> in\u00a0I<\/em> such that\u00a0P<\/em>(j<\/em>) holds. \u00a0(In other words, the set of indices j<\/em> such that\u00a0P<\/em>(j<\/em>) holds is cofinal in\u00a0I<\/em>.)<\/p>\n

If the index set were\u00a0N<\/strong> instead of\u00a0I<\/em>, that would be equivalent to saying that\u00a0P<\/em>(j<\/em>) holds for infinitely many values of\u00a0j<\/em>, if that can be of some help.<\/p>\n

We will also say\u00a0“P<\/em>(j<\/em>) holds for cofinally many\u00a0j\u2265i<\/em>” (or the obvious variants) to say that\u00a0for every\u00a0k<\/em> in\u00a0I<\/em>\u00a0with k<\/em>\u2265i<\/em>,\u00a0there is a\u00a0j<\/em>\u2265k<\/em> in\u00a0I<\/em> such that\u00a0P<\/em>(j<\/em>) holds. \u00a0If the index set were\u00a0N<\/strong>, that would mean that P<\/em>(j<\/em>) holds for infinitely many\u00a0j\u2265i<\/em>\u2014and the `\u2265i<\/em>‘ part would be useless.<\/p>\n

Lemma.<\/strong> For every\u00a0i<\/em> in\u00a0I<\/em>, and every closed subset C<\/em>\u00a0of Zi<\/sub><\/em>, if pij<\/sub><\/em>[Zj<\/sub><\/em>] intersects\u00a0C<\/em>\u00a0for cofinally many\u00a0j<\/em>\u2265i<\/em>, then\u00a0C<\/em>=Zi<\/sub><\/em>.<\/p>\n

Remark. \u00a0If\u00a0pij<\/sub><\/em>[Zj<\/sub><\/em>] intersects\u00a0C<\/em>\u00a0for cofinally many\u00a0j<\/em>\u2265i<\/em>, then\u00a0pij<\/sub><\/em>[Zj<\/sub><\/em>] intersects\u00a0C<\/em>\u00a0for every<\/em>\u00a0j<\/em>\u2265i<\/em>. \u00a0(Indeed, take any\u00a0j<\/em>\u2265i<\/em>. \u00a0Then there is a further\u00a0k<\/em>\u2265j<\/em> such that\u00a0pik<\/sub><\/em>[Zk<\/sub><\/em>] intersects\u00a0C<\/em>, by cofinality. \u00a0Hence there is a point\u00a0x<\/em> in\u00a0Zk<\/sub><\/em>\u00a0such that\u00a0pik<\/sub><\/em>(xk<\/sub><\/em>) is in\u00a0C<\/em>, and then\u00a0pjk<\/sub><\/em>(xk<\/sub><\/em>) is a point of\u00a0Zj<\/sub><\/em>\u00a0whose image by\u00a0pij<\/sub><\/em>\u00a0is in\u00a0C<\/em>.) \u00a0But we will really need the “cofinally many” part later.<\/p>\n

Proof.<\/em> \u00a0We build a new family of closed subset Z’k<\/sub><\/em>\u00a0of each Xk<\/sub><\/em>, as follows. \u00a0By the Remark, for every\u00a0j<\/em>\u2265i,k<\/em>,\u00a0pij<\/sub><\/em>[Zj<\/sub><\/em>] intersects\u00a0C<\/em>. \u00a0The set pij<\/sub><\/em>[Zj<\/sub><\/em>]\u00a0is included in Xj<\/sub><\/em>, but the fact that it intersects\u00a0C<\/em> can be expressed by saying that,\u00a0equivalently,\u00a0pij<\/sub><\/em>-1<\/sup>(C<\/em>) \u2229 Zj<\/sub><\/em> is non-empty. \u00a0Note that the set\u00a0pij<\/sub><\/em>-1<\/sup>(C<\/em>) \u2229 Zj<\/sub><\/em>\u00a0is included in Xi<\/sub><\/em>\u00a0instead.<\/p>\n

We can project back pij<\/sub><\/em>-1<\/sup>(C<\/em>) \u2229 Zj<\/sub><\/em>\u00a0onto a closed subset of Xk<\/sub><\/em>, by taking the closure cl (pkj<\/sub><\/em>[pij<\/sub><\/em>-1<\/sup>(C<\/em>) \u2229 Zj<\/sub><\/em>]) of its image by pkj<\/sub><\/em>. \u00a0This is a familiar trick with directed families (here,\u00a0I<\/em>): to go from index\u00a0i<\/em> to index\u00a0k<\/em>, where\u00a0k<\/em> is possibly incomparable to\u00a0i<\/em>, we find an index\u00a0j<\/em> above both\u00a0i<\/em> and\u00a0k<\/em>, then we go from\u00a0i<\/em> to\u00a0j<\/em>, and then back from\u00a0j<\/em> to\u00a0k<\/em>.<\/p>\n

One checks easily that the family F<\/em>\u00a0of closed sets\u00a0cl (pkj<\/sub><\/em>[pij<\/sub><\/em>-1<\/sup>(C<\/em>) \u2229 Zj<\/sub><\/em>]) where\u00a0j<\/em> ranges over the indices that are above both\u00a0i<\/em> and k<\/em>\u00a0is filtered: as j<\/em> grows, the sets\u00a0cl (pkj<\/sub><\/em>[pij<\/sub><\/em>-1<\/sup>(C<\/em>) \u2229 Zj<\/sub><\/em>]) become smaller and smaller. \u00a0Since\u00a0F<\/em> is a filtered family of non-empty closed sets in a compact space, its intersection is closed and non-empty: this is what we choose to be Z’k<\/sub><\/em>.<\/p>\n

It is also elementary to check that if\u00a0k<\/em>\u2264k’<\/em>, then pkk’<\/sub><\/em>\u00a0maps every element of Z’k’<\/sub><\/em>\u00a0to Z’k<\/sub><\/em>. \u00a0(Exercise. \u00a0Think of using the fact that\u00a0pkk’<\/sub><\/em>\u00a0is continuous, so the image of a closure is included in the closure of the image.) \u00a0Therefore\u00a0(Z’k<\/sub><\/em>)k\u2208 I<\/sub><\/em>\u00a0is an element of \u03a6.<\/p>\n

By construction of Z’k<\/sub><\/em>, Z’k\u00a0<\/sub><\/em>is included in\u00a0cl (pkj<\/sub><\/em>[pij<\/sub><\/em>-1<\/sup>(C<\/em>) \u2229 Zj<\/sub><\/em>]) for at least one\u00a0j<\/em>, hence in\u00a0cl (pkj<\/sub><\/em>[Zj<\/sub><\/em>])\u00a0\u2286 cl (Zk<\/sub><\/em>) = Zk<\/sub><\/em>. By maximality of (Zk<\/sub><\/em>)k\u2208 I<\/sub><\/em>, Z’k<\/sub><\/em>=Zk<\/sub><\/em> for every k<\/em> in I<\/em>.<\/p>\n

In particular, Z’i<\/sub><\/em>=Zi<\/sub><\/em>. \u00a0But \u00a0\u00a0Z’i\u00a0<\/sub><\/em>\u2286\u00a0cl (pij<\/sub><\/em>[pij<\/sub><\/em>-1<\/sup>(C<\/em>) \u2229 Zj<\/sub><\/em>]) for at least one\u00a0j<\/em>, and that is included in\u00a0cl (pij<\/sub><\/em>[pij<\/sub><\/em>-1<\/sup>(C<\/em>)])\u00a0\u2286\u00a0C<\/em>. \u00a0Therefore\u00a0Z’i\u00a0<\/sub><\/em>\u2286\u00a0C<\/em>. \u00a0Equality follows since\u00a0C<\/em> is a subset of Z’i<\/sub><\/em>. \u00a0\u00a0\u2610<\/p>\n

Zi<\/sub><\/em>\u00a0is irreducible closed<\/h2>\n

Using the Lemma, we can now proceed and show that Zi<\/sub><\/em>\u00a0is irreducible closed for every\u00a0i<\/em>. \u00a0Imagine that\u00a0Zi<\/sub><\/em>\u00a0is included in the union of two closed subsets\u00a0C’<\/em> and\u00a0C”<\/em> of Xi<\/sub><\/em>. \u00a0We wish to show that\u00a0Zi<\/sub><\/em>\u00a0is included in\u00a0C’<\/em> or in\u00a0C”<\/em>, namely that\u00a0Zi<\/sub><\/em>\u00a0\u2229 C’<\/em> or\u00a0Zi<\/sub><\/em>\u00a0\u2229 C”<\/em>\u00a0equals Zi<\/sub><\/em>. \u00a0To do so, we use the Lemma with\u00a0C<\/em> equal to\u00a0Zi<\/sub><\/em>\u00a0\u2229 C’<\/em> or to\u00a0Zi<\/sub><\/em>\u00a0\u2229 C”<\/em>.<\/p>\n

Explicitly, let\u00a0J’<\/em> be the set of indices\u00a0j<\/em>\u2265i<\/em> such that\u00a0pij<\/sub><\/em>[Zj<\/sub><\/em>] intersects\u00a0Zi<\/sub>\u00a0\u2229<\/em> C’<\/em>, and let\u00a0J”<\/em> be the set of indices\u00a0j<\/em>\u2265i<\/em> such that\u00a0pij<\/sub><\/em>[Zj<\/sub><\/em>] intersects\u00a0Zi<\/sub>\u00a0\u2229<\/em> C”<\/em>: we must show that\u00a0J’<\/em> or\u00a0J”<\/em> is cofinal in\u00a0I<\/em>.<\/p>\n

To do so, we claim that\u00a0J’<\/em>\u00a0\u222a J”<\/em> is cofinal in\u00a0I<\/em>. \u00a0In fact every\u00a0j<\/em>\u2265i<\/em> is in J’<\/em>\u00a0\u222a J”<\/em>. Indeed, for every\u00a0j<\/em>\u2265i<\/em>, Zj<\/sub><\/em>\u00a0is non-empty, so there is a point\u00a0x<\/em> in Zj<\/sub><\/em>, and\u00a0pij<\/sub><\/em>(x<\/i>) is in Zi<\/sub><\/em>. \u00a0If\u00a0pij<\/sub><\/em>(x<\/i>)\u00a0is in\u00a0C’<\/em> then\u00a0j<\/em> is in\u00a0J’<\/em>, otherwise\u00a0pij<\/sub><\/em>(x<\/i>) is in\u00a0C”<\/em> and\u00a0j<\/em> is in\u00a0J”<\/em>.<\/p>\n

Now, since\u00a0J’<\/em>\u00a0\u222a J”<\/em> is cofinal in\u00a0I<\/em>,\u00a0J’<\/em> or\u00a0J”<\/em> must be cofinal in\u00a0I<\/em>: otherwise after a certain rank no element of\u00a0I<\/em> would be in\u00a0J’<\/em>, and no element of\u00a0I<\/em> would be in\u00a0J”<\/em>. \u00a0This concludes the proof: summing up, if\u00a0J’<\/em> is cofinal in\u00a0I<\/em>, then we apply the Lemma with\u00a0C<\/em>=Zi<\/sub>\u00a0\u2229<\/em> C’<\/em>, so that\u00a0Zi<\/sub><\/em> \u2286\u00a0C’<\/em>, and if\u00a0J”<\/em> is cofinal in\u00a0I<\/em>, then we apply the Lemma with\u00a0C<\/em>=Zi<\/sub>\u00a0\u2229<\/em> C”<\/em>, so that\u00a0Zi<\/sub><\/em> \u2286\u00a0C”<\/em>. \u00a0\u2610<\/p>\n

Finishing the proof of the Proposition<\/h2>\n

Since every\u00a0Xi<\/sub><\/em>\u00a0is sober, the irreducible closed subset\u00a0Zi<\/sub><\/em>\u00a0is the closure \u2193xi<\/sub><\/em> of a unique point xi<\/sub><\/em>. \u00a0By the definition of\u00a0\u03a6,\u00a0pij<\/sub><\/em>\u00a0maps every point of Zj<\/sub><\/em>, in particular xj<\/sub><\/em>, to a point of Zi<\/sub><\/em>=\u2193xi<\/sub><\/em>, so\u00a0pij<\/sub><\/em>(xj<\/sub><\/em>)\u2264xi<\/sub><\/em>.<\/p>\n

In order to show the converse inequality, we recall from the proof of the Lemma that Z’i<\/sub><\/em>=Zi<\/sub><\/em>. \u00a0In particular xi<\/sub><\/em>\u00a0is in Z’i<\/sub><\/em>, hence in\u00a0cl (pij<\/sub><\/em>[pij<\/sub><\/em>-1<\/sup>(C<\/em>) \u2229 Zj<\/sub><\/em>])\u00a0\u2286\u00a0cl (pij<\/sub><\/em>[Zj<\/sub><\/em>]) =\u00a0cl (pij<\/sub><\/em>[\u2193x<\/em>j<\/sub><\/em>]). \u00a0Since pij<\/sub><\/em>\u00a0is continuous, hence monotonic,\u00a0pij<\/sub><\/em>[\u2193x<\/em>j<\/sub><\/em>]\u00a0\u2286\u00a0\u2193pij<\/sub><\/em>(x<\/em>j<\/sub><\/em>). \u00a0Therefore\u00a0xi<\/sub><\/em>\u00a0is in cl (\u2193pij<\/sub><\/em>(x<\/em>j<\/sub><\/em>)) =\u00a0\u2193pij<\/sub><\/em>(x<\/em>j<\/sub><\/em>). \u00a0This shows that\u00a0x<\/em>i<\/sub><\/em>\u2264pij<\/sub><\/em>(x<\/em>j<\/sub><\/em>), hence x<\/em>i<\/sub><\/em>=pij<\/sub><\/em>(x<\/em>j<\/sub><\/em>).<\/p>\n

As this holds for all\u00a0i<\/em>\u2264j<\/em>, the tuple\u00a0(xi<\/sub><\/em>)i\u2208 I<\/sub><\/em>\u00a0is in the projective limit\u00a0X<\/em>. \u00a0So\u00a0X<\/em> is non-empty. \u00a0This concludes the proof of the Proposition.\u00a0 \u2610<\/p>\n

That is enough for this month. \u00a0We have only proved that the projective limit of non-empty compact sober spaces is non-empty. \u00a0We have done the most complicated part! \u00a0Next month, we will see that this entails that projective limits of compact sober spaces are compact. \u00a0This requires much less effort.<\/p>\n

    \n
  1. Steenrod, Norman E. 1936. Universal Homology Groups. American Journal of Mathematics, 58(4), 661\u2013701.<\/li>\n
  2. Stone, Arthur Harold. 1979. Inverse Limits of Compact Spaces. General Topology and its Applications, 10, 203\u2013211.<\/li>\n
  3. Fujiwara, Kazuhiro, and Kato, Fumiharu. 2017 (Feb.). Foundations of Rigid Geometry I. arXiv 1308.4734<\/a>, v5.<\/li>\n<\/ol>\n

    \u2014 Jean Goubault-Larrecq<\/a>\u00a0(September 23rd, 2018)\"jgl-2011\"<\/p>\n\n\n

    <\/p>\n","protected":false},"excerpt":{"rendered":"

    Last time, I explained some of the strange things that happen with projective limits of topological spaces: they can be empty, even if all the spaces in the given projective system are non-empty and all bonding maps are surjective, and … 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":{"footnotes":""},"class_list":["post-1548","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1548","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=1548"}],"version-history":[{"count":14,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1548\/revisions"}],"predecessor-version":[{"id":5925,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1548\/revisions\/5925"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1548"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}

It is easy to see that given any directed family ((Cj<\/sup>i<\/sub><\/em>)i\u2208 I<\/sub><\/em>)j\u2208 J<\/sub><\/em> of elements of \u03a6 (namely, for each i<\/em> in\u00a0J<\/em>, the collection of sets (Cj<\/sup>i<\/sub><\/em>)j\u2208 J<\/sub><\/em>\u00a0is filtered, since we are working with reverse<\/em> inclusion), its pointwise intersection, (Ci<\/sub><\/em>)i\u2208 I<\/sub><\/em>, where each Ci<\/sub><\/em>\u00a0is defined as \u2229j\u2208 J<\/sub><\/em> Cj<\/sup>i<\/sub><\/em>, is again an element of \u03a6. The key point is that Ci<\/sub><\/em> is non-empty, because every filtered intersection of non-empty closed subsets of a compact space is non-empty (Exercise 4.4.11 in the book<\/a>.)<\/p>\n