{"id":964,"date":"2016-08-25T11:51:17","date_gmt":"2016-08-25T09:51:17","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=964"},"modified":"2022-11-19T15:24:09","modified_gmt":"2022-11-19T14:24:09","slug":"the-dolecki-greco-lechicki-theorem","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=964","title":{"rendered":"The Dolecki-Greco-Lechicki Theorem"},"content":{"rendered":"

I have met Szymon Dolecki<\/a> at the Summer Topology Conference 2016<\/a>, and he is a charming person.\u00a0 He tried to make me discover the best Sauvignon Blanc in the world, from the Marlborough vineyards in New Zealand; but I don’t drink wine, to his great disappointment.<\/p>\n

Apart from that, but still in relation with Szymon, Jimmie Lawson<\/a> mentioned the Dolecki-Greco-Lechicki Theorem [1, Theorem 4.1], a very nice theorem that attracts my attention more and more.\u00a0 This says that every \u010cech-complete space is consonant.\u00a0 (I’ll explain below.)<\/p>\n

That Theorem is the topic of Exercise 8.3.4 in the book<\/a>, but I am afraid that, as stated, it is way too hard. My purpose here is to give a complete solution to the exercise.<\/p>\n

Consonance<\/h2>\n

Consider the lattice O<\/strong>(X<\/em>) of all open subsets of a topological space X<\/em>. There are several ways we can topologize this.<\/p>\n

The Scott topology.<\/h3>\n

One of the most obvious ways, at least to domain theorists, is to give it the Scott topology (of the inclusion ordering).<\/p>\n

Szymon made several historical remarks as to who invented this topology first. If we look at convergence with respect to the Scott topology, then on O<\/strong>(X<\/em>), or rather on the anti-isomorphic lattice H<\/strong>(X<\/em>) of closed subsets of X<\/em>, Kazimierz Kuratowski had invented the same notion of convergence (of closed sets) a long time before Scott, and Szymon told us that Paul Painlev\u00e9 had invented it even earlier, and that Giuseppe Peano had invented it even earlier again, in 1887 \u2014 at a time when Kuratowski was not even born.<\/p>\n

He also mentioned what an extraordinary person Painlev\u00e9 was, managing to be, at the same time, a renowned scientist and a prominent politician. He was one of the leading mathematicians of his time, and worked mostly in the classification of solutions to second-order ordinary differential equations; in 1903, he showed that the equation of fluid mechanics made flying possible, and later completely developed the theory of flight in perfect fluids. He was elected a member of the French Academy of Sciences, and eventually became its president. At the same time, he got interested in politics in the course of the infamous Dreyfus affair, and ended up being several times minister of war, several times “pr\u00e9sident du conseil” (=prime minister), and several times minister of the (newly created) air force, between the two world wars.<\/p>\n

By the way, Peano was also one extraordinary mathematician, and Szymon kindly directed me to his paper [2], with Gabriele Greco: Peano had invented convergence notions for sets, in Euclidean space, well before Kuratowski, Painlev\u00e9, but also Hausdorff or Vietoris.\u00a0 I discovered they had a subsequent paper<\/a>, too, on the “amazing oblivion of Peano’s contributions to mathematics”.\u00a0 Szymon tells me this has evolved into a series of three papers called “The astonishing oblivion of Peano’s mathematical legacy<\/i>“, available from his preprints page<\/a>, and which should make it into a little book.\u00a0 It is, indeed, amazing how much of modern mathematics Peano had invented, which was later rediscovered by Borel, Lebesgue, Zermelo, Banach, Choquet, and others.<\/p>\n

The compact-open topology.<\/h3>\n

Anyway, there is at least one other natural topology on O<\/strong>(X<\/em>). Since O<\/strong>(X<\/em>) can be identified with the space of continuous maps from X<\/em> to Sierpi\u0144ski space S<\/strong>, one can equip it with the compact-open topology. (One can also equip it with the Isbell topology, for example, but in that case the Isbell topology is<\/em> the Scott topology.) This topology is generated by basic open sets of the form \uffedQ<\/em>, where Q<\/em> is compact saturated in X<\/em>.\u00a0 By definition, \uffedQ<\/em> is the collection of open neighborhoods of Q<\/em>.<\/p>\n

In other words, a set U<\/strong><\/em> of open subsets of X<\/em> is open in the compact-open topology if and only if, for every U<\/em> in U<\/strong><\/em>, there is a compact saturated subset Q<\/em> of X<\/em> such that U<\/em> \u2208 \uffedQ<\/em> \u2286 U<\/strong><\/em>.<\/p>\n

It is fairly easy to see that \uffedQ<\/em> is always Scott-open.\u00a0 Therefore every open in the compact-open topology is also open in the Scott topology.<\/p>\n

Dolecki, Greco and Lechicki [1] call consonant<\/em> those spaces X<\/em> for which the converse also holds, i.e., those for which the Scott and the compact-open topologies agree on O<\/strong>(X<\/em>).\u00a0 Concretely, those are the spaces X<\/em> such that, if you take any Scott-open subset U<\/strong><\/em> of O<\/strong>(X<\/em>), and any U<\/em> in U<\/strong><\/em>, then there is a compact saturated subset Q<\/em> of X<\/em> contained in U<\/strong><\/em> such that every open neighborhood of Q<\/em> is in U<\/strong><\/em>.<\/p>\n

The following results are part of Exercise 5.4.12 in the book<\/a>:<\/p>\n