{"id":1559,"date":"2018-10-23T19:46:33","date_gmt":"2018-10-23T17:46:33","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1559"},"modified":"2022-11-19T15:15:33","modified_gmt":"2022-11-19T14:15:33","slug":"project-limits-of-topological-spaces-iii-finishing-the-proof-of-steenrods-theorem","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1559","title":{"rendered":"Projective limits of topological spaces III: finishing the proof of Steenrod’s theorem"},"content":{"rendered":"

Last time, we had started to prove the following theorem [3, Theorem 2.2.20]:<\/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

My goal this time is to finish its proof. \u00a0That is unfortunately pretty technical. \u00a0If there were anything to remember from the proof, that would be that the following Proposition, which we had proved last time, is the cornerstone of the proof.<\/p>\n

Proposition.\u00a0<\/strong> Let\u00a0 (p’jk\u00a0<\/sub><\/em>: Ck<\/sub><\/em>\u00a0\u2192 Cj<\/sub><\/em>)j\u2264k\u2208J<\/sub><\/em>\u00a0be a projective system of compact sober spaces. \u00a0If every\u00a0Cj<\/sub><\/em>\u00a0is non-empty, then its projective limit\u00a0is non-empty as well.<\/p>\n

The Proposition of course only deals with the final part of the Theorem. \u00a0Showing that the projective limit is sober is easy, and we deal with that point next. \u00a0Showing that it is compact is much more technical, and we will repeatedly use the Proposition to prove that. \u00a0The general idea will be to find non-empty closed (hence compact and sober) subsets Ci<\/sub><\/em>\u00a0of each Xi<\/sub><\/em>, in such a way as to define a new projective limit\u00a0(pij\u00a0<\/sub><\/em>: Cj<\/sub><\/em>\u00a0\u2192 Ci<\/sub><\/em>)i\u2264j \u2208 I<\/sub><\/em> of compact sober subspaces, and to use the fact that this projective limit is non-empty in order to make progress.<\/p>\n

Sobriety<\/h2>\n

Not all subspaces of a sober space are sober: take any non-sober space\u00a0X<\/em>, and realize that it occurs (up to homeomorphism) as a subspace of its sobrification. \u00a0However, any subspace [g<\/em>1<\/sub>=g<\/em>2<\/sub>] of X<\/em>\u00a0that is built as an equalizer of two continuous maps\u00a0g<\/em>1<\/sub>, g<\/em>2<\/sub> : X<\/em>\u00a0\u2192\u00a0Y<\/em> with\u00a0X<\/em> sober (Lemma 8.4.12 of the book<\/a>; recall that, in an explicit form,\u00a0[g<\/em>1<\/sub>=g<\/em>2<\/sub>] is the subspace of all\u00a0x<\/em> in\u00a0X<\/em> such that\u00a0g<\/em>1<\/sub>(x<\/em>)=g<\/em>2<\/sub>(x<\/em>).)<\/p>\n

This shows immediately that:<\/p>\n

Lemma 1.<\/strong> \u00a0A projective limit X<\/em>\u00a0of a projective system (pij\u00a0<\/sub><\/em>: Xj<\/sub><\/em>\u00a0\u2192 Xi<\/sub><\/em>)i\u2264j \u2208 I<\/sub><\/em>\u00a0of sober spaces is sober.<\/p>\n

This is because\u00a0\u03a0i\u2208 I<\/sub><\/em> Xi<\/sub><\/em>\u00a0is sober (Theorem 8.4.8 in the book<\/a>) and\u00a0X<\/em> occurs as the equalizer\u00a0[g<\/em>1<\/sub>=g<\/em>2<\/sub>] where\u00a0g<\/em>1<\/sub>\u00a0:\u00a0\u03a0i\u2208 I<\/sub><\/em> Xi<\/sub><\/em>\u00a0\u2192 \u03a0i\u2264j\u2208 I<\/sub><\/em> Xi<\/sub><\/em>\u00a0maps\u00a0(xi<\/sub><\/em>)i\u2208 I<\/sub><\/em>\u00a0to\u00a0(pij<\/sub><\/em>(xj<\/sub><\/em>))i\u2264j\u2208 I<\/sub><\/em>\u00a0and\u00a0g<\/em>2<\/sub>\u00a0maps\u00a0(xi<\/sub><\/em>)i\u2208 I<\/sub><\/em>\u00a0to\u00a0(xi<\/sub><\/em>)i\u2264j\u2208 I<\/sub><\/em>. \u00a0(Please pay attention to the fact that the target space of\u00a0g<\/em>1\u00a0<\/sub>and\u00a0g<\/em>2<\/sub>\u00a0is a product indexed, not by\u00a0i<\/em>, but by all pairs of indices\u00a0i<\/em>,\u00a0j<\/em> such that i<\/em>\u2264j<\/em>.)<\/p>\n

This also shows, for example, that every open subspace\u00a0U<\/em>\u00a0of a sober space\u00a0X<\/em> is sober, being the equalizer of the characteristic map of\u00a0U<\/em> (with values in Sierpi\u0144ski space\u00a0S<\/strong>) with the constant 1 map.<\/p>\n

Every closed subspace\u00a0C<\/em> is also sober, as the equalizer of the characteristic map of its complement with the constant 0 map.<\/p>\n

A compactness lemma<\/h2>\n

Let\u00a0X<\/em> be the projective limit of a projective system of compact sober spaces\u00a0(pij\u00a0<\/sub><\/em>: Xj<\/sub><\/em>\u00a0\u2192 Xi<\/sub><\/em>)i\u2264j \u2208 I<\/sub><\/em>. \u00a0\u00a0Let also\u00a0pi<\/sub><\/em>\u00a0:\u00a0X<\/em>\u00a0\u2192 Xi\u00a0<\/sub><\/em>be the projection maps. \u00a0We have the following, which one can think as a sort of compactness lemma for the set of indices\u00a0I<\/em>.<\/p>\n

We will not have any use of it in this post, but this is a generally useful lemma, and it can be taken as a gentle example of how we can make good use of the Proposition we mentioned at the beginning of the post (the projective limit of a projective system of non-empty compact sober spaces is non-empty).<\/p>\n

Lemma 2.<\/strong>\u00a0 For every\u00a0i<\/em> in\u00a0I<\/em>, and every open neighborhood\u00a0V<\/em> of the image of X<\/em> by\u00a0pi<\/sub><\/em>, there is an index\u00a0j<\/em>\u2265i<\/em> in\u00a0I<\/em> such that that\u00a0V<\/em> already contains the image of Xj<\/sub><\/em>\u00a0by\u00a0pij<\/sub><\/em>.<\/p>\n

Proof.<\/em>\u00a0\u00a0We reason by contradiction, and assume that\u00a0V<\/em> does not contain the image of Xj<\/sub><\/em>\u00a0by\u00a0pij<\/sub><\/em>\u00a0for any\u00a0j<\/em>\u2265i<\/em>. \u00a0Let J<\/em>\u00a0be the subset of those indices of\u00a0I<\/em> above\u00a0i<\/em>. \u00a0This is a directed set. \u00a0For each\u00a0j<\/em> in\u00a0J<\/em>,\u00a0Cj<\/sub><\/em>=Xj<\/sub><\/em>\u2013pij<\/sub><\/em>-1<\/sup>(V<\/em>) is then non-empty. \u00a0It is closed in a compact space, hence is itself compact. \u00a0As a compact subset, it is also a compact subspace<\/em>\u00a0(Exercise 4.9.11 of the book<\/a>). \u00a0As a closed subspace of a sober space, it is sober.<\/p>\n

For all\u00a0j<\/em>\u2264k<\/em> in\u00a0J<\/em>, the restriction p’jk<\/sub><\/em>\u00a0of pjk<\/sub><\/em>\u00a0to Ck<\/sub><\/em>\u00a0maps every element to an element of\u00a0Cj<\/sub><\/em>. \u00a0Indeed, for every element\u00a0x<\/em> of Ck<\/sub><\/em>, if pjk<\/sub><\/em>(x<\/em>) were in\u00a0pij<\/sub><\/em>-1<\/sup>(V<\/em>), then pij<\/sub><\/em>(pjk<\/sub><\/em>(x<\/em>))=pik<\/sub><\/em>(x<\/em>) would be in\u00a0V<\/em>, and\u00a0x<\/em> would be in\u00a0pik<\/sub><\/em>-1<\/sup>(V<\/em>): contradiction.<\/p>\n

Therefore\u00a0 (p’jk\u00a0<\/sub><\/em>: Ck<\/sub><\/em>\u00a0\u2192 Cj<\/sub><\/em>)j\u2264k\u2208J<\/sub><\/em>\u00a0is a projective system of non-empty compact sober spaces. \u00a0The Proposition we mentioned at the beginning of the post shows that it has a non-empty projective limit. \u00a0We pick a tuple (xj<\/sub><\/em>)j\u2208 J<\/sub><\/em>\u00a0in that projective limit. \u00a0We can complete it to a tuple indexed by\u00a0I<\/em> instead, defining xj<\/sub><\/em>\u00a0for\u00a0j<\/em> not in\u00a0J<\/em> as\u00a0pjk<\/sub><\/em>(xk<\/sub><\/em>) for some arbitrary\u00a0k<\/em> in\u00a0I<\/em> above both\u00a0i<\/em> and\u00a0j<\/em>. \u00a0This defines an element\u00a0x<\/em> of\u00a0X<\/em> such that\u00a0pj<\/sub><\/em>(x<\/em>)=xj<\/sub><\/em> for every\u00a0j\u00a0<\/em>in\u00a0J<\/em>, in particular pi<\/sub><\/em>(x<\/em>)=xi<\/sub><\/em>. \u00a0It follows that xi<\/sub><\/em>\u00a0is in the image of pi<\/sub><\/em>, hence in\u00a0V<\/em>, and that is impossible since\u00a0xi<\/sub><\/em>\u00a0is in Ci<\/sub><\/em>. \u2610<\/p>\n

Compactness<\/h2>\n

In order to show the Theorem, we need to show that the projective limit\u00a0X<\/em> of a projective system of compact sober spaces\u00a0(pij\u00a0<\/sub><\/em>: Xj<\/sub><\/em>\u00a0\u2192 Xi<\/sub><\/em>)i\u2264j \u2208 I<\/sub><\/em>\u00a0is compact. \u00a0\u00a0Let also\u00a0pi<\/sub><\/em>\u00a0:\u00a0X<\/em>\u00a0\u2192 Xi\u00a0<\/sub><\/em>be the projection maps, as usual. \u00a0That is the hard part, and I hope you will not be lost in indices. There will be quite a lot of them. \u00a0In the sequel,\u00a0i<\/em> and\u00a0j<\/em> will always denote indices from the index set\u00a0I<\/em>, while\u00a0k<\/em> will index the open subsets from a directed open cover of\u00a0X<\/em>.<\/p>\n

In order to show that X<\/em> is compact, we consider a directed family (Uk<\/sub><\/em>)k \u2208 K<\/sub><\/em>\u00a0of open subsets of\u00a0X<\/em>, whose union is equal to\u00a0X<\/em>, and we wish to show that some Uk<\/sub><\/em>\u00a0is already equal to the whole of\u00a0X<\/em>. \u00a0(This is Proposition 4.4.7 of the book<\/a>. \u00a0Being able to consider the family as directed will simplify the argument.)<\/p>\n

For each\u00a0k<\/em> in\u00a0K<\/em>, and every i<\/em> in\u00a0I<\/em>,\u00a0there is a largest open subset Uki<\/sub><\/em>\u00a0of\u00a0Xi\u00a0<\/sub><\/em>such that\u00a0pi<\/sub><\/em>-1<\/sup>(Uki<\/sub><\/em>)\u00a0\u2286 Uk<\/sub><\/em>: we just take the union of all open subsets\u00a0U<\/em> of\u00a0Xi\u00a0<\/sub><\/em>such that\u00a0pi<\/sub><\/em>-1<\/sup>(U<\/em>)\u00a0\u2286 Uk<\/sub><\/em>.<\/p>\n

We notice that as i<\/em> grows,\u00a0pi<\/sub><\/em>-1<\/sup>(Uki<\/sub><\/em>) becomes larger. \u00a0(Formal argument: If\u00a0i<\/em>\u2264j<\/em>, then pi<\/sub><\/em>= pij<\/sub><\/em>\u00a0o pj<\/sub><\/em>, pi<\/sub><\/em>-1<\/sup>(Uki<\/sub><\/em>) is equal to\u00a0pj<\/sub><\/em>-1<\/sup>(pij<\/sub><\/em>-1<\/sup>(Uki<\/sub><\/em>)). \u00a0This shows that\u00a0pij<\/sub><\/em>-1<\/sup>(Uki<\/sub><\/em>) is an open subset\u00a0U<\/em> of\u00a0Xj\u00a0<\/sub><\/em>such that\u00a0pj<\/sub><\/em>-1<\/sup>(U<\/em>)\u00a0\u2286 Uk<\/sub><\/em>. \u00a0It must then be included in the largest one,\u00a0Ukj<\/sub><\/em>. \u00a0So\u00a0pi<\/sub><\/em>-1<\/sup>(Uki<\/sub><\/em>)\u00a0\u2286\u00a0pj<\/sub><\/em>-1<\/sup>(Ukj<\/sub><\/em>).) \u00a0Hence the family of sets\u00a0pi<\/sub><\/em>-1<\/sup>(Uki<\/sub><\/em>), when\u00a0i<\/em> varies, is directed. \u00a0Its union is included in Uk<\/sub><\/em>, and we claim that:<\/p>\n

Lemma 3.<\/strong> \u00a0The directed union of the sets pi<\/sub><\/em>-1<\/sup>(Uki<\/sub><\/em>), i<\/em>\u2208I<\/em>, is equal to Uk<\/sub><\/em>.<\/p>\n

Proof.<\/em> \u00a0Consider any point\u00a0(xi<\/sub><\/em>)i\u2208 I<\/sub><\/em>\u00a0in Uk<\/sub><\/em>. \u00a0Since Uk<\/sub><\/em>\u00a0is open, and remembering that the topology on\u00a0X<\/em> is the subspace topology from a product topology, there is a finite subset\u00a0J<\/em> of\u00a0I<\/em> and open subsets Vi\u00a0<\/sub><\/em>of Xi<\/sub><\/em>,\u00a0i<\/em>\u00a0\u2208\u00a0J<\/em>, such that xi<\/sub><\/em> \u2208\u00a0Vi<\/sub><\/em> for every\u00a0i<\/em>\u00a0\u2208\u00a0J<\/em>, and such that every tuple whose\u00a0i<\/em>th coordinate is in\u00a0Vi\u00a0<\/sub><\/em>for every\u00a0i<\/em>\u00a0\u2208\u00a0J<\/em>, is in\u00a0Uk<\/sub><\/em>. \u00a0Since\u00a0I<\/em> is directed, there is a an index\u00a0j<\/em> in\u00a0I<\/em> above every element of J<\/em>. \u00a0For every\u00a0i<\/em>\u00a0\u2208\u00a0J<\/em>, pij<\/sub><\/em>(xj<\/sub><\/em>)=xi<\/sub><\/em> is in Vi<\/sub><\/em>, so xj<\/sub><\/em>\u00a0is in\u00a0\u2229i\u2208J<\/sub><\/em> pij<\/sub><\/em>-1<\/sup>(Vi<\/sub><\/em>). The open set\u00a0pj<\/sub><\/em>-1<\/sup>(\u2229i\u2208J<\/sub><\/em> pij<\/sub><\/em>-1<\/sup>(Vi<\/sub><\/em>)) consists of tuples (yi<\/sub><\/em>)i\u2208 I<\/sub><\/em>\u00a0such that\u00a0yj<\/sub><\/em>\u00a0is in\u00a0\u2229i\u2208J<\/sub><\/em> pij<\/sub><\/em>-1<\/sup>(Vi<\/sub><\/em>), namely such that yi\u00a0<\/sub><\/em>is in Vi<\/sub><\/em>\u00a0for every\u00a0i<\/em>\u00a0\u2208\u00a0J<\/em>, and is therefore included in Uk<\/sub><\/em>. \u00a0By the maximality property of Ukj<\/sub><\/em>, \u2229i\u2208J<\/sub><\/em> pij<\/sub><\/em>-1<\/sup>(Vi<\/sub><\/em>)\u00a0is included in Ukj<\/sub><\/em>. \u00a0Therefore\u00a0xj<\/sub><\/em>\u00a0is in Ukj<\/sub><\/em>. \u00a0It follows that\u00a0(xi<\/sub><\/em>)i\u2208 I<\/sub><\/em>\u00a0is in\u00a0pj<\/sub><\/em>-1<\/sup>(Ukj<\/sub><\/em>), showing the claim. \u00a0\u2610<\/p>\n

Since X<\/em> is the directed union of the sets Uk<\/sub><\/em>,\u00a0k<\/em>\u2208K<\/em>,\u00a0X<\/em> is also the directed union over\u00a0k<\/em> of the directed union over\u00a0i<\/em> of the sets\u00a0pi<\/sub><\/em>-1<\/sup>(Uki<\/sub><\/em>). Switching the two unions,\u00a0X<\/em> is also the directed union over\u00a0i<\/em> of the open sets Vi<\/sub><\/em>, where Vi<\/sub><\/em>\u00a0is defined as the (directed) union of (Uki<\/sub><\/em>)k\u2208K<\/sub><\/em>.<\/p>\n

Note that\u00a0Vi<\/sub><\/em>\u00a0is an open subset of the compact set Xi<\/sub><\/em>, and is defined as a directed union of open subsets. \u00a0This sounds good, but the latter is not (yet known to be) an open\u00a0cover<\/em>, so we cannot conclude (yet).<\/p>\n

Instead, we define Ci<\/sub><\/em>\u00a0as the complement of Vi<\/sub><\/em>\u00a0in Xi<\/sub><\/em>. \u00a0This is a closed subset of Xi<\/sub><\/em>, and as in the proof of Lemma 2,\u00a0Ci<\/sub><\/em>\u00a0is a compact sober subspace of Xi<\/sub><\/em>.<\/p>\n

For all\u00a0i<\/em>\u2264j<\/em> in\u00a0I<\/em>,\u00a0pij<\/sub><\/em>-1<\/sup>(Uki<\/sub><\/em>) is included in Ukj<\/sub><\/em>. \u00a0(Formal argument: It suffices to check that\u00a0pj<\/sub><\/em>-1<\/sup>(pij<\/sub><\/em>-1<\/sup>(Uki<\/sub><\/em>)), which is equal to\u00a0pi<\/sub><\/em>-1<\/sup>(Uki<\/sub><\/em>), is included in Uk<\/sub><\/em>, and to invoke the maximality of Uki<\/sub><\/em>.) \u00a0Taking unions over\u00a0k<\/em> in\u00a0K<\/em>,\u00a0pij<\/sub><\/em>-1<\/sup>(Vi<\/sub><\/em>) is included in Vj<\/sub><\/em>. \u00a0Hence the image of Cj<\/sub><\/em>\u00a0by pij<\/sub><\/em>\u00a0is included in Ci<\/sub><\/em>: if there were a point of Cj\u00a0<\/sub><\/em>whose image by pij<\/sub><\/em>\u00a0were not in Ci<\/sub><\/em>, hence in Vi<\/sub><\/em>, it would be in\u00a0pij<\/sub><\/em>-1<\/sup>(Vi<\/sub><\/em>) and therefore in Vj<\/sub><\/em>, contradiction.<\/p>\n

All this means that\u00a0(p’ij\u00a0<\/sub><\/em>: Cj<\/sub><\/em>\u00a0\u2192 Ci<\/sub><\/em>)i\u2264j \u2208 I<\/sub><\/em>\u00a0is a(nother) projective system of compact sober spaces, where p’ij\u00a0<\/sub><\/em>is the restriction of pij<\/sub><\/em>\u00a0to Cj<\/sub><\/em>.<\/p>\n

But that projective system has an empty projective limit! \u00a0Let us check that.
\nImagine there were an element its its projective limit. \u00a0That element would be in\u00a0pi<\/sub><\/em>-1<\/sup>(Ci<\/sub><\/em>) for every\u00a0i<\/em> in\u00a0I<\/em>, hence not in\u00a0pi<\/sub><\/em>-1<\/sup>(Vi<\/sub><\/em>) for any\u00a0i<\/em> in\u00a0I<\/em>, but that is impossible since\u00a0X<\/em> is the union over all\u00a0i<\/em> in\u00a0I<\/em> over \u00a0pi<\/sub><\/em>-1<\/sup>(Vi<\/sub><\/em>).<\/p>\n

The Proposition mentioned at the beginning of this post (projective limits of non-empty compact sober spaces are non-empty) tells us that this cannot happen if every Ci<\/sub><\/em>\u00a0is non-empty. \u00a0Hence\u00a0Ci<\/sub><\/em>\u00a0is empty for some\u00a0i<\/em> in\u00a0I<\/em>, and this means that\u00a0Vi<\/sub><\/em>=Xi<\/sub><\/em>.<\/p>\n

Since\u00a0Vi<\/sub><\/em>\u00a0is the directed union of (Uki<\/sub><\/em>)k\u2208K<\/sub><\/em>, and Xi<\/sub><\/em>\u00a0is compact,\u00a0Xi<\/sub><\/em>=Uki<\/sub><\/em> for some\u00a0k<\/em> in\u00a0K<\/em>. \u00a0Finally, every element of\u00a0the projective limit\u00a0X<\/em> is in\u00a0pi<\/sub><\/em>-1<\/sup>(Xi<\/sub><\/em>) (meaning that it\u00a0i<\/em>th coordinate must be in Xi<\/sub><\/em>), namely in\u00a0pi<\/sub><\/em>-1<\/sup>(Uki<\/sub><\/em>), which is included in\u00a0Uk<\/sub><\/em>. \u00a0Hence\u00a0X<\/em>=Uk<\/sub><\/em>.\u00a0 \u2610<\/p>\n

That finishes the proof of the Theorem.<\/p>\n

Instead of ending this post here, let me conclude with an easy observation on\u00a0sobrifications of projective limits.<\/p>\n

Sobrifications of projective limits<\/h2>\n

Remember that the sobrification functor S<\/strong>\u00a0preserves products (Theorem 8.4.8 in the book<\/a>), so we may think that it would perhaps preserve all limits in\u00a0Top<\/strong>. \u00a0That is wrong, since it does not preserve equalizers: see the warning comments after Lemma 8.4.12 in the book<\/a>.<\/p>\n

We will now argue that\u00a0S<\/strong> does not preserve projective limits in general\u00a0either.<\/p>\n

Recall the following counterexample, due to A. H. Stone, and mentioned as Example 3 in my first post on projective limits<\/a>. \u00a0For every n<\/em> \u2208 N<\/strong>, we define\u00a0Xn<\/sub><\/em>\u00a0as\u00a0N<\/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>. \u00a0The bonding maps\u00a0pmn\u00a0<\/sub><\/em>: Xn<\/sub><\/em> \u2192Xm<\/sub><\/em>\u00a0(m<\/em>\u2264n<\/em>) are identity maps.<\/p>\n

We had seen that every\u00a0Xn<\/sub><\/em> is compact, in fact Noetherian, and T1<\/sub>. \u00a0The projective limit X<\/em>\u00a0is\u00a0N<\/strong> with the discrete<\/em> topology, and we had observed that this is not compact.<\/p>\n

Let us look at the irreducible closed subsets C<\/em>\u00a0of Xn<\/sub><\/em>. \u00a0The only ones that are finite are the one-element sets. \u00a0It remains to look at those such that C<\/em> \u2229 \u2191n<\/em>=\u2191n<\/em>. \u00a0Those are obtained as\u00a0\u2191n<\/em> union finitely many elements below\u00a0n\u2013<\/em>1. \u00a0Irreducibility implies there cannot be any element\u00a0below\u00a0n\u2013<\/em>1, hence\u00a0C<\/em>=\u2191n<\/em>. \u00a0Conversely, it is easy to see that\u00a0\u2191n<\/em> is irreducible closed in Xn<\/sub><\/em>.<\/p>\n

It follows that the sobrification S<\/strong>(Xn<\/sub><\/em>)\u00a0of Xn<\/sub><\/em>\u00a0is obtained by adding a fresh element \u03c9n\u00a0<\/sub><\/em>(corresponding to \u2191n<\/em>), above\u00a0n<\/em>,\u00a0n<\/em>+1, …, but incomparable with 0, 1, …,\u00a0n<\/em>\u20131.<\/p>\n

Since sobrification is a functor, the projective system\u00a0(pmn\u00a0<\/sub><\/em>: Xn<\/sub><\/em>\u00a0\u2192 Xm<\/sub><\/em>)m\u2264n \u2208 <\/sub><\/em>N<\/strong><\/sub>\u00a0gives rise to a new projective system\u00a0(S<\/strong>(pmn<\/sub><\/em>): S<\/strong>(Xn<\/sub><\/em>) \u2192 S<\/strong>(Xm<\/sub><\/em>))m\u2264n \u2208 <\/sub><\/em>N<\/strong><\/sub>. \u00a0The sobrification of a compact space is compact, since it has an isomorphic lattice of open subsets. \u00a0Hence we have obtained a \u00a0projective system of compact sober spaces, and Steenrod’s theorem tells us that its projective limit X’<\/em>\u00a0is compact and sober.<\/p>\n

It follows that\u00a0X’<\/em> cannot be the sobrification of\u00a0X<\/em>, which is not compact. \u00a0So the sobrification functor does not preserve projective limits.<\/p>\n

Explicitly,\u00a0X<\/em> is\u00a0N<\/strong> with the discrete topology, hence is already sober. \u00a0X’<\/em>, instead, is\u00a0N<\/strong> plus an extra element\u00a0\u03c9, incomparable with all natural numbers, and is indeed different from X.<\/em><\/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(October 23rd, 2018)\"jgl-2011\"<\/p>\n","protected":false},"excerpt":{"rendered":"

    Last time, we had started to prove the following theorem [3, Theorem 2.2.20]: Theorem. The projective limit of a projective system (pij\u00a0: Xj\u00a0\u2192 Xi)i\u2264j \u2208 I of compact sober spaces is compact and sober. It is non-empty if every Xi … 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-1559","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1559","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=1559"}],"version-history":[{"count":12,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1559\/revisions"}],"predecessor-version":[{"id":5924,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1559\/revisions\/5924"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1559"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}