{"id":6007,"date":"2022-12-19T22:15:08","date_gmt":"2022-12-19T21:15:08","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=6007"},"modified":"2023-01-08T11:54:13","modified_gmt":"2023-01-08T10:54:13","slug":"aliaume-lopez-master-theorem-of-noetherian-spaces","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=6007","title":{"rendered":"Aliaume Lopez’ master theorem of Noetherian spaces"},"content":{"rendered":"\n

First of all, Season’s Greetings!<\/p>\n\n\n\n

Second, I would like to mention that this blog has now been in existence for 10 years now, too. The first post<\/a> dates from November 5th, 2012, and was a succinct summary of Matthew de Brecht’s quasi-Polish spaces. Another reason to celebrate!<\/p>\n\n\n

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\"\"<\/a><\/figure>\n<\/div>\n\n\n

But enough talking; let us get to today’s topic.<\/p>\n\n\n\n

In the theory of any kind of spaces, it is useful to have ways of constructing spaces. For example, you may want to know that products of bc-domains, that spaces of Scott-continuous functions between bc-domains, all are bc-domains.<\/p>\n\n\n\n

In the case of Noetherian spaces, I have given a few constructions in Section 9.7 of the <\/a>book<\/a>: the Hoare hyperspace of a Noetherian space is Noetherian, finite products and finite coproducts of Noetherian spaces are Noetherian, the space of finite words over a Noetherian alphabet is Noetherian, the space of finite trees over a Noetherian signature is Noetherian, etc. All of them are proved in pretty different ways, and one may wonder whether there would be some kind of master theorem that would include all, or at least most, of the constructions at once.<\/p>\n\n\n\n

For the purpose of this post, let me say (or remind you) that a space X<\/em> is Noetherian if and only if every monotone sequence of open sets (Un<\/sub><\/em>)n<\/em> \u2208 N<\/strong><\/sub> stabilizes, namely: there is an index n<\/em>0<\/sub> such that all the sets Un<\/sub><\/em> with n<\/em>\u2265n<\/em>0<\/sub> are equal. Equivalently, X<\/em> is Noetherian if and only if every open subset of X<\/em> is compact. I will also say that a topology<\/em> \u03c4 on a set X<\/em> is Noetherian if and only if the space (X<\/em>, \u03c4) is Noetherian.<\/p>\n\n\n\n

What Aliaume Lopez achieved<\/h2>\n\n\n\n

Aliaume Lopez<\/a> proved such a theorem this year [1]. He found his proof by scrutinizing the proof of Theorem 9.7.46 in the\u00a0<\/a>book<\/a> (the topological Kruskal theorem, on finite trees), and distilling its essence. The result is pretty remarkable. There have been pretty similar theorems for well-quasi-orders [2,3], and Aliaume shows how Ryu Hasegawa’s construction [2], for one, is a consequence of his theorem; his construction goes further, not just because it applies to the larger class of Noetherian spaces, but also because it shows the Noetherianity of spaces that are not obtained as initial constructions, contrarily to [2,3].<\/p>\n\n\n\n

What he does is as follows. Imagine you start with some Noetherian topology on a set X<\/em>, and that you have an operator E<\/em> that maps Noetherian topologies on X<\/em> to Noetherian topologies. Now iterate E<\/em>, possibly transfinitely, and take the topology generated by the union of all the iterates. This gives you a new topology, which I will call the limit topology<\/em> of E<\/em>. When is that limit topology Noetherian?<\/p>\n\n\n\n

(Oh, before you ask: that limit topology has nothing to do with categorical limits or colimits, in any sense that I know of.)<\/p>\n\n\n\n

Aliaume discovered that the crucial property that is needed in order to achieve this is that E<\/em> should be what he calls a topology expander<\/em>. This is defined by a few conditions. The first one is pretty innocuous: E<\/em> should be monotonic, in other words, it should map finer Noetherian topologies to finer Noetherian topologies. The important condition is pretty mysterious, and I will come back to it later.<\/p>\n\n\n\n

Oh well, I am lying slightly, too. Aliaume considers maps E<\/em> from all<\/em> topologies to topologies, which the proviso that E<\/em> respects Noetherianness. But we do not need to define E<\/em> on non-Noetherian topologies.<\/p>\n\n\n\n

Meanwhile, let us consider an example. This is also the example that Aliaume uses to motivate the construction [1, Definition 3.9, Lemma 3.10]. I modified it slightly; the result is a bit simpler in my view.<\/p>\n\n\n\n

The example of words<\/h2>\n\n\n\n

Let me say, or recall, how the word topology<\/em> on the set X<\/em>* of finite words on Noetherian space X<\/em> is constructed (see Definition 9.7.26 in the <\/a>book<\/a>).<\/p>\n\n\n\n

The elements of X<\/em>* are the words<\/em> over the alphabet X<\/em>. Let us call a subword<\/em> of a word w<\/em> any word obtained as a not necessarily contiguous subsequence of the letters of w<\/em>. For example, cd<\/i>e<\/em> is a subword of abcdefg<\/em>, but bdeg<\/em> is a subword as well. (This is not the word embedding preordering \u2264* defined after Definition 9.7.26 in the <\/a>book<\/a>. Aliaume uses \u2264*, but we will obtain similar results with the simpler notion of subwords.) For a more complicated example involving several occurrences of the same variable, aab<\/i> is a subword of acacbc<\/em>.<\/p>\n\n\n\n

The word topology on X<\/em>* has a base of open sets of the form X<\/em>*U<\/em>1<\/sub>X<\/em>*…X<\/em>*U<\/em>n<\/em><\/sub>X<\/em>*, where U<\/em>1<\/sub>, …, U<\/em>n<\/em><\/sub> are arbitrary open subsets of X<\/em>. Another way of writing these basic open sets is as ⬆U<\/em>1<\/sub>…U<\/em>n<\/em><\/sub>, where:<\/p>\n\n\n\n