{"id":2702,"date":"2020-08-23T10:53:39","date_gmt":"2020-08-23T08:53:39","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=2702"},"modified":"2022-11-19T15:02:06","modified_gmt":"2022-11-19T14:02:06","slug":"chains-and-nested-spaces","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=2702","title":{"rendered":"Chains and nested spaces"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">A&nbsp;<em>nested space<\/em>&nbsp;is any topological space in which the lattice of open sets is totally ordered. &nbsp;My purpose this month is to show that this concept, which perhaps looks strange at first:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>is related to the familiar concept of a chain (a totally ordered set) in order theory, and that chains and nested spaces have very strong topological properties;<\/li>\n\n\n\n<li>is related to the notion of minimal&nbsp;T<sub>0&nbsp;<\/sub>and T<sub><em>D<\/em>&nbsp;<\/sub>topologies, a concept first introduced by Larson [1].<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">I will start with a nifty, and perhaps surprising, observation about chains.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">A nifty observation about chains<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A chain is a totally ordered set. Can you find one that is not continuous, as a poset? Well, do not think too much. There is none. If I remember well, the following was told to me by Xiaodong Jia.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Proposition A.<\/strong>&nbsp;Let&nbsp;<em>P<\/em>&nbsp;be a chain, and &lt; be its strictly smaller than relation. Then: (i) if&nbsp;<em>x<\/em>&lt;<em>y<\/em>&nbsp;then&nbsp;<em>x<\/em>\u226a<em>y<\/em>; (ii)&nbsp;<em>P<\/em>&nbsp;is a continuous poset.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Proof.<\/em>&nbsp;(i) Let&nbsp;<em>D<\/em>&nbsp;be any directed family with a supremum sup&nbsp;<em>D<\/em>&nbsp;such that&nbsp;<em>y<\/em>\u2264sup&nbsp;<em>D<\/em>. (Note that &#8220;directed&#8221; is almost entirely irrelevant, here, by the way: every non-empty family in&nbsp;<em>P<\/em>&nbsp;is automatically directed.) We claim that&nbsp;<em>x<\/em>\u2264<em>d<\/em>&nbsp;for some element&nbsp;<em>d<\/em>&nbsp;of&nbsp;<em>D<\/em>. What would happen if that were not the case? Well, using the totality of \u2264, for every&nbsp;<em>d<\/em>&nbsp;in&nbsp;<em>D<\/em>, we would have&nbsp;<em>d<\/em>&lt;<em>x<\/em>, in particular&nbsp;<em>d<\/em>\u2264<em>x<\/em>. Then sup&nbsp;<em>D<\/em>\u2264<em>x<\/em>, which is impossible since&nbsp;<em>x<\/em>&lt;<em>y<\/em>\u2264sup&nbsp;<em>D<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">(ii) Let&nbsp;<em>x<\/em>&nbsp;be any point of&nbsp;<em>P<\/em>. If&nbsp;<em>x<\/em>&nbsp;is finite, namely if&nbsp;<em>x<\/em>\u226a<em>x<\/em>, then {<em>x<\/em>} is a directed family of elements way-below&nbsp;<em>x<\/em>&nbsp;whose supremum is&nbsp;<em>x<\/em>. Otherwise,&nbsp;<em>x<\/em>&nbsp;is not finite, so by definition there is a directed family&nbsp;<em>D<\/em>&nbsp;with a supremum, such that&nbsp;<em>x<\/em>\u2264sup&nbsp;<em>D<\/em>, but for every&nbsp;<em>d<\/em>&nbsp;in&nbsp;<em>D<\/em>, it is not the case that&nbsp;<em>x<\/em>\u2264<em>d<\/em>. Using totality, for every&nbsp;<em>d<\/em>&nbsp;in&nbsp;<em>D<\/em>,&nbsp;<em>d<\/em>&lt;<em>x<\/em>. By (i), for every&nbsp;<em>d<\/em>&nbsp;in&nbsp;<em>D<\/em>,&nbsp;<em>d<\/em>\u226a<em>x<\/em>, so&nbsp;<em>D<\/em>&nbsp;is a family of elements way-below&nbsp;<em>x<\/em>.&nbsp;<em>D<\/em>&nbsp;is also directed, since in a chain every non-empty set is directed. It remains to note that sup&nbsp;<em>D<\/em>=<em>x<\/em>. Since&nbsp;<em>d<\/em>&lt;<em>x<\/em>&nbsp;for every&nbsp;<em>d<\/em>&nbsp;in&nbsp;<em>D<\/em>, we have sup&nbsp;<em>D<\/em>\u2264<em>x<\/em>, and we already know that&nbsp;<em>x<\/em>\u2264sup&nbsp;<em>D<\/em>. \u2610<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If&nbsp;<em>P<\/em>&nbsp;is not only a chain, but also a pointed dcpo, then we have even more. &nbsp;Remember that a poset is pointed if and only if it has a least element \u22a5. &nbsp;First, every family at all has a supremum: if that family is empty, then the supremum is \u22a5, otherwise the family is directed\u2026 because&nbsp;<em>P<\/em>&nbsp;is a chain. &nbsp;Hence&nbsp;<em>P<\/em>&nbsp;is a complete lattice.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let me also recall that, for any two points\u00a0<em>x<\/em>\u00a0and\u00a0<em>y<\/em>\u00a0of a complete lattice\u00a0<em>P<\/em>,\u00a0<em>x<\/em>\u00a0is\u00a0<em>way-way-below y<\/em>\u00a0(in notation,\u00a0<em>x<\/em>\u22d8<em>y<\/em>), if and only if for every family\u00a0<em>D<\/em>\u00a0(not necessarily directed) such that\u00a0<em>y<\/em>\u2264sup\u00a0<em>D<\/em>, some element of\u00a0<em>D<\/em>\u00a0is already above\u00a0<em>x<\/em>. \u00a0A complete lattice is\u00a0<em>prime-continuous<\/em>\u00a0if and only if every element\u00a0<em>x\u00a0<\/em>is the supremum of the elements way-way-below\u00a0<em>x<\/em>. \u00a0And Raney\u2019s theorem (Exercise 8.3.16 in the <a href=\"https:\/\/www.cambridge.org\/gb\/knowledge\/isbn\/item7069109\/Non-Hausdorff%20Topology%20and%20Domain%20Theory\/?site_locale=en_GB\">book<\/a>) states that a complete lattice is prime-continuous if and only if it is completely distributive.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If&nbsp;<em>P<\/em>&nbsp;is a chain and a pointed dcpo, we have&nbsp;<em>x<\/em>\u22d8<em>y<\/em>&nbsp;if and only if&nbsp;<em>x<\/em><em>\u226a<\/em><em>y<\/em>&nbsp;and&nbsp;<em>y<\/em>\u2260\u22a5 (the latter is dictated by the case where&nbsp;<em>D<\/em>&nbsp;is empty). &nbsp;Every element different from&nbsp;\u22a5 is then the (directed) supremum of a family of elements way-below (hence way-way-below) itself, by Proposition A. &nbsp;And&nbsp;\u22a5 itself is the supremum of the empty family. &nbsp;Hence we obtain the following.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Corollary B.<\/strong>&nbsp;&nbsp;Every chain that is also a pointed dcpo is a completely distributive complete lattice. &nbsp;\u2610<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Nested spaces<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Can we find a topological analogue of the notion of a chain? Surely yes. That is the notion of a&nbsp;<em>nested space<\/em>: a space&nbsp;<em>X<\/em> whose lattice of open sets is a chain, in other words, such that given any two open sets&nbsp;<em>U<\/em>&nbsp;and&nbsp;<em>V<\/em>, one of them is included in the other.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">That may seem like a strange condition, but look: every chain is nested in its Scott topology (and also, in its Alexandroff topology, and in its upper topology).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If&nbsp;<em>X<\/em>&nbsp;is a nested space, then&nbsp;<strong>O<\/strong><em>X<\/em>&nbsp;is a chain, hence is a continuous complete lattice. We know that the spaces whose lattices of open sets are continuous (Definition 5.2.3) are exactly the core-compact spaces, so every nested space is core-compact.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">But we can say much more. \u00a0<strong>O<em>X<\/em><\/strong>\u00a0is not only a chain, but also a pointed dcpo, so Corollary B applies:\u00a0<strong>O<em>X<\/em><\/strong>\u00a0is a completely distributive complete lattice. \u00a0This suggests that\u00a0<em>X<\/em>\u00a0is a c-space, as suggested by Lemma 8.3.42 of the <a href=\"https:\/\/www.cambridge.org\/gb\/knowledge\/isbn\/item7069109\/Non-Hausdorff%20Topology%20and%20Domain%20Theory\/?site_locale=en_GB\">book<\/a>\u2026 which however requires\u00a0<em>X<\/em>\u00a0to be sober. \u00a0We confirm this intuition, and give a direct proof of it.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Proposition C.<\/strong>&nbsp;Every nested space is a c-space.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let me recall that a c-space is a space in which for every point&nbsp;<em>x<\/em>, for every open neighborhood&nbsp;<em>U<\/em>&nbsp;of&nbsp;<em>x<\/em>, one can find a point&nbsp;<em>y<\/em>&nbsp;such that&nbsp;<em>x<\/em>&nbsp;\u2208 int(\u2191<em>y<\/em>) \u2286 \u2191<em>y<\/em>&nbsp;\u2286&nbsp;<em>U<\/em>. Being a c-space is a much stronger condition than being core-compact.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Proof.<\/em>&nbsp;&nbsp;Let&nbsp;<em>X<\/em>&nbsp;be a nested space, and let \u2264 be its specialization preordering. We first claim that \u2264 is total, namely that for any pair of points&nbsp;<em>x<\/em>,&nbsp;<em>y<\/em>, we have&nbsp;<em>x<\/em>\u2264<em>y<\/em>&nbsp;or&nbsp;<em>y<\/em>\u2264<em>x<\/em>. Let us imagine that&nbsp;<em>x<\/em>\u2270<em>y<\/em>. Then there is an open neighborhood&nbsp;<em>U<\/em>&nbsp;of&nbsp;<em>x<\/em>&nbsp;that does not contain&nbsp;<em>y<\/em>. For every open neighborhood&nbsp;<em>V<\/em>&nbsp;of&nbsp;<em>y<\/em>, we cannot have&nbsp;<em>V<\/em>&nbsp;\u2286&nbsp;<em>U<\/em>, since that would force&nbsp;<em>U<\/em>&nbsp;to contain&nbsp;<em>y<\/em>. By nestedness,&nbsp;<em>U<\/em>&nbsp;is included in&nbsp;<em>V<\/em>. Since&nbsp;<em>U<\/em>&nbsp;contains&nbsp;<em>x<\/em>,&nbsp;<em>V<\/em>&nbsp;must also contain&nbsp;<em>x<\/em>. This shows that&nbsp;<em>y<\/em>\u2264<em>x<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Next, we claim that for every point&nbsp;<em>y<\/em>, the set \u2191<sub>&lt;<\/sub>&nbsp;<em>y<\/em>&nbsp;of all the points&nbsp;<em>x<\/em>&nbsp;such that&nbsp;<em>y<\/em>&lt;<em>x<\/em>&nbsp;is open. Oh, I write&nbsp;<em>y<\/em>&lt;<em>x<\/em>&nbsp;for &#8220;<em>y<\/em>\u2264<em>x<\/em>&nbsp;and&nbsp;<em>x<\/em>\u2270<em>y<\/em>&#8220;. Since \u2264 is a preordering, not necessarily an ordering, this is not equivalent to &#8220;<em>y<\/em>\u2264<em>x<\/em>&nbsp;and&nbsp;<em>x<\/em>\u2260<em>y<\/em>&#8220;. However, since \u2264 is total,&nbsp;<em>y<\/em>&lt;<em>x<\/em>&nbsp;actually simplifies to &#8220;<em>x<\/em>\u2270<em>y<\/em>&#8220;. Therefore \u2191<sub>&lt;<\/sub>&nbsp;<em>y<\/em>&nbsp;is just the complement of \u2193<em>y<\/em>, which is the closure of {<em>y<\/em>}, so indeed \u2191<sub>&lt;<\/sub>&nbsp;<em>y<\/em>&nbsp;is open.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">As a third point, we make a detour through the following situation. Let us imagine an open subset&nbsp;<em>U<\/em>&nbsp;of&nbsp;<em>X<\/em>&nbsp;with a minimal point&nbsp;<em>x<\/em>. By minimal, I mean that <em>x<\/em> is in <em>U<\/em>, but all the points&nbsp;<em>y<\/em>&lt;<em>x<\/em>&nbsp;are outside&nbsp;<em>U<\/em>. I claim that, in this case,&nbsp;<em>U<\/em>=\u2191<em>x<\/em>. That \u2191<em>x<\/em>&nbsp;is included in&nbsp;<em>U<\/em>&nbsp;is clear, since&nbsp;<em>U<\/em>&nbsp;is upwards-closed. In the converse direction, any point&nbsp;<em>y<\/em>&nbsp;in&nbsp;<em>U<\/em>&nbsp;that is not in \u2191<em>x<\/em>&nbsp;satisfies&nbsp;<em>y<\/em>&lt;<em>x<\/em>, since \u2264 is total, and this is impossible.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We can now show that&nbsp;<em>X<\/em>&nbsp;is a c-space. Let&nbsp;<em>x<\/em>&nbsp;be a point of&nbsp;<em>X<\/em>, and&nbsp;<em>U<\/em>&nbsp;be an open neighborhood of&nbsp;<em>x<\/em>. If&nbsp;<em>x<\/em>&nbsp;is minimal in&nbsp;<em>U<\/em>, then we have seen that&nbsp;<em>U<\/em>=\u2191<em>x<\/em>. Hence, taking&nbsp;<em>y<\/em>=<em>x<\/em>, we obtain&nbsp;<em>x<\/em>&nbsp;\u2208 int(\u2191<em>y<\/em>) \u2286 \u2191<em>y<\/em>&nbsp;\u2286&nbsp;<em>U<\/em>&nbsp;rather vacuously. Otherwise, there is a point&nbsp;<em>y<\/em>&lt;<em>x<\/em>&nbsp;in&nbsp;<em>U<\/em>. Then&nbsp;<em>x<\/em>&nbsp;is in \u2191<sub>&lt;<\/sub>&nbsp;<em>y<\/em>, which is open, as we have seen earlier, hence included in int(\u2191<em>y<\/em>). Finally, \u2191<em>y<\/em>&nbsp;\u2286&nbsp;<em>U<\/em>&nbsp;since&nbsp;<em>y<\/em>&nbsp;is in&nbsp;<em>U<\/em>. \u2610<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">I have said that every chain is a nested space in its Scott, Alexandroff, or upper topology. &nbsp;We can say something much more precise than that! &nbsp;The following is Lemma 2 of [1], up to some rephrasing.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Lemma D.<\/strong>&nbsp;Let&nbsp;<em>X<\/em>&nbsp;be a T<sub>0&nbsp;<\/sub>space. &nbsp;<em>X<\/em>&nbsp;is nested if and only if&nbsp;<em>X<\/em>&nbsp;is a chain in its specialization ordering.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Proof.<\/em>&nbsp;Assume that&nbsp;<em>X<\/em>&nbsp;is not a chain in its specialization ordering. &nbsp;Then it has two incomparable points,&nbsp;<em>x<\/em>&nbsp;and&nbsp;<em>y<\/em>. &nbsp;From&nbsp;<em>x\u2270y<\/em>&nbsp;we deduce that there is an open set&nbsp;<em>U<\/em>&nbsp;that contains&nbsp;<em>x<\/em>&nbsp;but not&nbsp;<em>y<\/em>. &nbsp;Symmetrically, there is an open set&nbsp;<em>V<\/em>&nbsp;that contains&nbsp;<em>y<\/em>&nbsp;but not&nbsp;<em>x<\/em>. &nbsp;Then&nbsp;<em>U<\/em>&nbsp;is not included in&nbsp;<em>V<\/em>, since&nbsp;<em>x<\/em>&nbsp;is in&nbsp;<em>U<\/em>&nbsp;but not in&nbsp;<em>V<\/em>, and&nbsp;<em>V<\/em>&nbsp;is not included in&nbsp;<em>U<\/em>, since&nbsp;<em>y<\/em>&nbsp;is in&nbsp;<em>V<\/em> but not in&nbsp;<em>U<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Conversely, assume that&nbsp;<em>X<\/em>&nbsp;is not nested. &nbsp;It has two incomparable open subsets&nbsp;<em>U<\/em>&nbsp;and&nbsp;<em>V<\/em>. &nbsp;Since&nbsp;<em>U<\/em>&nbsp;is not included in&nbsp;<em>V<\/em>, there is a point&nbsp;<em>x<\/em>&nbsp;in&nbsp;<em>U<\/em>&nbsp;that is not in&nbsp;<em>V<\/em>. &nbsp;Symmetrically, there is a point&nbsp;<em>y<\/em>&nbsp;in&nbsp;<em>V<\/em>&nbsp;that is not in&nbsp;<em>U<\/em>. &nbsp;Since&nbsp;<em>x<\/em>&nbsp;is in&nbsp;<em>U<\/em>&nbsp;and&nbsp;<em>y<\/em>&nbsp;is not in&nbsp;<em>U<\/em>,&nbsp;<em>x<\/em><em>\u2270y<\/em>. &nbsp;Since&nbsp;<em>y<\/em>&nbsp;is in&nbsp;<em>V<\/em>, and&nbsp;<em>x<\/em>&nbsp;is not in&nbsp;<em>V<\/em>,&nbsp;<em>y<\/em><em>\u2270<\/em><em>x<\/em>. \u2610<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Minimal T<sub>0<\/sub>&nbsp;topologies<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">If you start from an ordering \u2264 on a set&nbsp;<em>P<\/em>, it is a standard result, due to Szpilrajn [2], that (the graph of) \u2264 is included in (the graph of) some total ordering \u2aaf. In other words, \u2264 has a&nbsp;<em>linearization<\/em>. In computer science, \u2aaf is known as a&nbsp;<em>topological sort<\/em>&nbsp;of \u2264. &nbsp;(Apparently, Szpilrajn remarked that&nbsp;Banach, Kuratowski, and Tarski already knew of the result, although none published it.)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The proof of this fact is an application of Zorn&#8217;s Lemma. The family of all orderings that contain \u2264 is a dcpo, hence an inductive poset, under inclusion. And every maximal ordering in that dcpo must be a total ordering. &nbsp;In order to prove the latter, given any ordering \u2264\u2019 for which&nbsp;<em>x<\/em>&nbsp;and&nbsp;<em>y<\/em>&nbsp;are incomparable points, we can build a new ordering \u2264\u2019\u2019 &nbsp;by&nbsp;<em>z<\/em>&nbsp;\u2264\u2019\u2019&nbsp;<em>t<\/em>&nbsp;if and only&nbsp;<em>z<\/em>&nbsp;\u2264\u2019&nbsp;<em>t<\/em>&nbsp;or (<em>z<\/em>&nbsp;\u2264\u2019&nbsp;<em>x<\/em>&nbsp;and&nbsp;<em>y<\/em>&nbsp;\u2264\u2019&nbsp;<em>t<\/em>), and \u2264\u2019\u2019 is strictly larger than \u2264\u2019. If \u2264\u2019 is maximal, then the existence of two incomparable points for \u2264\u2019 would then lead to a contradiction.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The topological analogue is the notion of a coarsest T<sub>0&nbsp;<\/sub>topology, usually called a&nbsp;<em>minimal T<sub>0<\/sub>&nbsp;topology<\/em>. Indeed, if you remove some open sets, then the specialization ordering (not preordering: this is why we only consider&nbsp;<em>T<sub>0<\/sub><\/em>&nbsp;topologies) grows under inclusion.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The notion of a minimal&nbsp;T<sub>0&nbsp;<\/sub>topology was first studied by R. E. Larson [1], as far as I know. &nbsp;He also studied minimal T<sub><em>D<\/em>&nbsp;<\/sub>topologies, which we will do next.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">I will proceed differently from Larson, in order to show the connection with linearizations of orderings (and I will mention Larson\u2019s simpler argument later). \u00a0The following is Theorem 1 of [1], more or less. \u00a0Let me recall that the upper topology of an ordering \u2264 is the coarsest topology such that \u2193<em>x<\/em>\u00a0is closed for every point\u00a0<em>x<\/em>. \u00a0Explicitly, its\u00a0closed sets are intersections of sets of the form \u2193<em>E<\/em>,\u00a0<em>E<\/em>\u00a0finite and non-empty. \u00a0(If \u2264 is total, then every such set \u2193<em>E<\/em>\u00a0can be written as \u2193<em>x<\/em>\u00a0for a unique point\u00a0<em>x<\/em>.) \u00a0The upper topology of \u2264 is the coarsest topology that has \u2264 as specialization (pre)ordering (see Proposition 4.2.12 in the <a href=\"https:\/\/www.cambridge.org\/gb\/knowledge\/isbn\/item7069109\/Non-Hausdorff%20Topology%20and%20Domain%20Theory\/?site_locale=en_GB\">book<\/a>).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Proposition E.<\/strong>&nbsp;&nbsp;The minimal T<sub>0&nbsp;<\/sub>topologies on any fixed set&nbsp;<em>X<\/em>&nbsp;are exactly the upper topologies of total orderings on&nbsp;<em>X<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Proof.<\/em>&nbsp; Let&nbsp;\u03c4\u2019 be a minimal T<sub>0&nbsp;<\/sub>topology on&nbsp;<em>X<\/em>.&nbsp; The upper topology of its specialization ordering \u2264\u2019 is even coarser. &nbsp;Minimality then implies that \u03c4\u2019 is the upper topology of \u2264\u2019.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Now let us imagine that \u2264\u2019 is not a total ordering. &nbsp;We have two points&nbsp;<em>x<\/em>&nbsp;and&nbsp;<em>y<\/em>&nbsp;of&nbsp;<em>X<\/em>&nbsp;such that&nbsp;<em>y<\/em>&nbsp;\u2270\u2019&nbsp;<em>x<\/em>&nbsp;and&nbsp;<em>x<\/em>&nbsp;\u2270\u2019&nbsp;<em>y<\/em>. &nbsp;As in the proof of Szpilrajn\u2019s theorem, we build a new ordering \u2264\u2019\u2019 &nbsp;by&nbsp;<em>z<\/em>&nbsp;\u2264\u2019\u2019&nbsp;<em>t<\/em>&nbsp;if and only&nbsp;<em>z<\/em>&nbsp;\u2264\u2019&nbsp;<em>t<\/em>&nbsp;or (<em>z<\/em>&nbsp;\u2264\u2019&nbsp;<em>x<\/em>&nbsp;and&nbsp;<em>y<\/em>&nbsp;\u2264\u2019&nbsp;<em>t<\/em>); then \u2264\u2019\u2019 is strictly larger than \u2264\u2019. &nbsp;The downward closure \u2193\u2019\u2019<em>t&nbsp;<\/em>of any point&nbsp;<em>t<\/em>&nbsp;with respect to \u2264\u2019\u2019 &nbsp;is equal to its downward closure \u2193\u2019<em>t&nbsp;<\/em>with respect to \u2264\u2019 if&nbsp;<em>y<\/em>&nbsp;\u2270\u2019&nbsp;<em>t<\/em>, and to \u2193\u2019<em>t&nbsp;<\/em>&nbsp;\u222a \u2193\u2019<em>x<\/em>&nbsp;if&nbsp;<em>y<\/em>&nbsp;\u2264\u2019&nbsp;<em>t<\/em>. &nbsp;In particular, whichever the case is, \u2193\u2019\u2019<em>t<\/em>&nbsp;is always closed in the upper topology of \u2264\u2019, namely in \u03c4\u2019. &nbsp;This shows that the upper topology of \u2264\u2019\u2019 is coarser than&nbsp;\u03c4\u2019. &nbsp;By minimality, those two topologies are equal. &nbsp;In particular, their specialization orderings coincide, namely \u2264\u2019\u2019 and \u2264\u2019 coincide: but this is impossible, since \u2264\u2019\u2019 is strictly larger than \u2264\u2019.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore \u2264\u2019 is a chain. &nbsp;This shows that any minimal T<sub>0&nbsp;<\/sub>topology \u03c4\u2019 is indeed the upper topology of some total ordering \u2264\u2019.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Conversely, let \u2264\u2019 be any total ordering, and&nbsp;\u03c4\u2019 be its upper topology. &nbsp;Let us imagine that there is a coarser T<sub>0&nbsp;<\/sub>topology&nbsp;\u03c4\u2019\u2019. &nbsp;Its specialization ordering \u2264\u2019\u2019 must contain \u2264\u2019. &nbsp;But any total ordering is maximal with respect to inclusion (among ordering relations, not preordering relations). &nbsp;Therefore \u2264\u2019\u2019 is equal to \u2264\u2019. &nbsp;Then, the upper topology of \u2264\u2019 is the coarsest topology whose specialization ordering is \u2264\u2019, so&nbsp;\u03c4\u2019 is coarser than&nbsp;\u03c4\u2019\u2019. &nbsp;It follows that&nbsp;\u03c4\u2019=\u03c4\u2019\u2019. &nbsp;This allows us to assert that \u03c4\u2019 is minimal T<sub>0&nbsp;<\/sub>. &nbsp;\u2610<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">One would expect to obtain an analogue of Szpilrajn\u2019s theorem, typically that every T<sub>0&nbsp;<\/sub>topology on a fixed set&nbsp;<em>X<\/em>&nbsp;is finer than some minimal T<sub>0&nbsp;<\/sub>topology. &nbsp;However, that is wrong. &nbsp;This is explained in the second part of Example 6 of [1]. &nbsp;Let me make that explicit.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Counterexample F.<\/strong>&nbsp; The cofinite topology on the real line&nbsp;<strong>R<\/strong>&nbsp;is not finer than any minimal T<sub>0&nbsp;<\/sub>topology.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Proof.<\/em>&nbsp;&nbsp;Imagine there were such a minimal T<sub>0&nbsp;<\/sub>topology&nbsp;\u03c4. &nbsp;It is the upper topology of some total ordering \u2264. &nbsp;Let us write \u2193<em>x<\/em>&nbsp;for the downward closure of&nbsp;<em>x<\/em>&nbsp;with respect to \u2264. &nbsp;Since&nbsp;\u03c4 is coarser than the cofinite topology, \u2193<em>x<\/em>&nbsp;must be either finite or the whole of&nbsp;<strong>R<\/strong>, for each real number&nbsp;<em>x<\/em>.&nbsp; Let us define #<em>x<\/em>&nbsp;as the cardinality of \u2193<em>x.<\/em>&nbsp;&nbsp;If&nbsp;<em>x<\/em>\u2264<em>y<\/em>&nbsp;then #<em>x<\/em>\u2264#<em>y<\/em>&nbsp;(with the usual ordering on&nbsp;<strong>N<\/strong>&nbsp;\u222a {\u221e}), and if&nbsp;<em>x<\/em>&lt;<em>y<\/em>&nbsp;then #<em>x<\/em>&nbsp;is strictly smaller than #<em>y<\/em>. &nbsp;In particular, the map&nbsp;<em>x<\/em>&nbsp;\u21a6 #<em>x<\/em>&nbsp;is an injective map from&nbsp;<strong>R<\/strong>&nbsp;to&nbsp;<strong>N<\/strong>&nbsp;\u222a {\u221e}. &nbsp;That is impossible, since&nbsp;<strong>R<\/strong>&nbsp;is uncountable. &nbsp;\u2610<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In Counterexample F, of course the cofinite topology is T<sub>0<\/sub>: it is even T<sub>1<\/sub>. &nbsp;(In fact, on any set, the cofinite topology is the&nbsp;<em>minimal<\/em>&nbsp;T<sub>1&nbsp;<\/sub>topology.)<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Minimal T<em><sub>D<\/sub><\/em>&nbsp;topologies<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Larson also studied minimal T<sub><em>D<\/em>&nbsp;<\/sub>topologies. &nbsp;I have talked about T<sub><em>D<\/em>&nbsp;<\/sub>topologies <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=2626\">last time<\/a> [3]. &nbsp;Those are the topologies in which \u2193<em>x<\/em>\u2013{<em>x<\/em>} is closed for every point&nbsp;<em>x<\/em>. &nbsp;Every such topology is T<sub>0<\/sub>. &nbsp;The Alexandroff topology of any partial ordering is T<sub><em>D<\/em><\/sub>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Larson proved that a T<sub><em>D<\/em>&nbsp;<\/sub>topology is minimal T<sub><em>D<\/em><\/sub>, that is, it is T<sub><em>D<\/em><\/sub>&nbsp;and no strictly coarser topology is T<sub><em>D<\/em><\/sub>, if only if it is nested [1, Theorem 2]. &nbsp;We can say more: they are all Alexandroff topologies.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Proposition E\u2019.<\/strong>&nbsp;&nbsp;The minimal T<sub><em>D<\/em>&nbsp;<\/sub>topologies on any fixed set&nbsp;<em>X<\/em>&nbsp;are exactly the Alexandroff topologies of total orderings on&nbsp;<em>X<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Proof.<\/em>&nbsp;We reuse Larson\u2019s construction. &nbsp;Given any topology&nbsp;\u03c4 on&nbsp;<em>X<\/em>, and any open subset&nbsp;<em>U<\/em>&nbsp;in&nbsp;\u03c4, let&nbsp;\u03c4* be {<em>V<\/em>&nbsp;\u2208&nbsp;\u03c4 |&nbsp;<em>V<\/em>&nbsp;\u2286&nbsp;<em>U<\/em>&nbsp;or&nbsp;<em>U<\/em>&nbsp;\u2286&nbsp;<em>V<\/em>}. &nbsp;It is easy to see that&nbsp;\u03c4* is a topology, and that it is coarser than&nbsp;\u03c4. &nbsp;Let&nbsp;<em>C<\/em>&nbsp;be the complement of&nbsp;<em>U<\/em>. &nbsp;Then the \u03c4*-closed sets are the&nbsp;\u03c4-closed sets&nbsp;<em>D<\/em>&nbsp;such that&nbsp;<em>D<\/em>&nbsp;\u2286&nbsp;<em>C<\/em>&nbsp;or&nbsp;<em>C<\/em>&nbsp;\u2286&nbsp;<em>D<\/em>. &nbsp;It follows that the&nbsp;\u03c4*-closure \u2193*<em>x&nbsp;<\/em>of any point&nbsp;<em>x<\/em>&nbsp;is its&nbsp;\u03c4-closure \u2193<em>x<\/em>&nbsp;if&nbsp;<em>x<\/em>&nbsp;\u2208&nbsp;<em>C<\/em>, otherwise&nbsp;it is <em>C<\/em>&nbsp;\u222a \u2193<em>x<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In particular, if&nbsp;\u03c4 is T<sub><em>D<\/em><\/sub>, then so is&nbsp;\u03c4*: if&nbsp;<em>x<\/em>&nbsp;\u2208&nbsp;<em>C<\/em>, then \u2193*<em>x<\/em>\u2013{<em>x<\/em>}=\u2193<em>x<\/em>\u2013{<em>x<\/em>} is&nbsp;\u03c4-closed and included in&nbsp;<em>C<\/em>, hence&nbsp;\u03c4*-closed; otherwise, \u2193*<em>x<\/em>\u2013{<em>x<\/em>}=<em>C<\/em>&nbsp;\u222a (\u2193<em>x<\/em>\u2013{<em>x<\/em>}), which is&nbsp;\u03c4-closed and contains&nbsp;<em>C<\/em>, hence is also&nbsp;\u03c4*-closed. &nbsp;(One can also show that if \u03c4 is T<sub>0<\/sub>, then so is&nbsp;\u03c4*, and this is how Larson proves Proposition E.)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If&nbsp;\u03c4 is minimal T<sub><em>D<\/em><\/sub>, then&nbsp;\u03c4* cannot be strictly coarser than&nbsp;\u03c4, so&nbsp;\u03c4*=\u03c4. &nbsp;This means that for every&nbsp;<em>V<\/em>&nbsp;\u2208&nbsp;\u03c4,&nbsp;<em>V<\/em>&nbsp;\u2286&nbsp;<em>U<\/em>&nbsp;or&nbsp;<em>U<\/em>&nbsp;\u2286&nbsp;<em>V<\/em>. &nbsp;Since&nbsp;<em>U<\/em>&nbsp;is arbitrary,&nbsp;\u03c4 is nested.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Finally, we show that&nbsp;\u03c4 is the Alexandroff topology of its specialization ordering \u2264. &nbsp;By Lemma D, \u2264 is total. &nbsp;Hence the upward closure \u2191<em>x<\/em>&nbsp;of any point&nbsp;<em>x<\/em>&nbsp;is also equal to the complement of \u2193<em>x<\/em>\u2013{<em>x<\/em>}, which is&nbsp;\u03c4-closed. &nbsp;In other words, \u2191<em>x<\/em>&nbsp;is open. &nbsp;Since every upward-closed set&nbsp;<em>A<\/em>&nbsp;is a union of sets of the form \u2191<em>x, x<\/em>&nbsp;\u2208&nbsp;<em>A<\/em>, it is open. &nbsp; \u2610<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The argument of Counterexample F immediately leads to the following similar counterexample, showing that again there is no analogue of Szpilrajn\u2019s theorem in the class of T<sub><em>D<\/em>&nbsp;<\/sub>topologies.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Counterexample F\u2019.<\/strong>&nbsp; The cofinite topology on the real line&nbsp;<strong>R<\/strong>&nbsp;is not finer than any minimal T<sub><em>D<\/em>&nbsp;<\/sub>topology. &nbsp; \u2610<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We note that the cofinite topology, by the way, is T<sub><em>D<\/em><\/sub>, because T<sub>1&nbsp;<\/sub>implies T<sub><em>D<\/em><\/sub>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Counterexamples F and F&#8217; are pretty depressing observations: they show that there is no topological generalization of Szpilrajn\u2019s (order-theoretic) theorem.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">I cannot finish on such a sad note, so let me mention a remarkable application of Corollary B.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Mislove&#8217;s observation<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A continuous valuation is an analogue of a measure. Formally, a valuation \u03bd on a topological space&nbsp;<em>X<\/em>&nbsp;is a map from the lattice&nbsp;<strong>O<\/strong><em>X<\/em>&nbsp;of open subsets of&nbsp;<em>X<\/em>&nbsp;to&nbsp;<strong><span style=\"text-decoration: underline;\">R<\/span><\/strong><sub>+<\/sub>&nbsp;(the dcpo of non-negative real numbers, plus \u221e) that is monotonic, strict (\u03bd(\u2205)=0), and modular (\u03bd(<em>U<\/em>)+\u03bd(<em>V<\/em>)=\u03bd(<em>U<\/em>&nbsp;\u222a&nbsp;<em>V<\/em>)+\u03bd(<em>U<\/em>&nbsp;\u2229&nbsp;<em>V<\/em>)). A&nbsp;<em>continuous<\/em>&nbsp;valuation is a valuation that is Scott-continuous.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The set&nbsp;<strong>V<em>X<\/em><\/strong>&nbsp;can be ordered by the&nbsp;<em>stochastic ordering<\/em>, whereby \u03bc\u2264\u03bd if and only if&nbsp;\u03bc(<em>U<\/em>)<em>\u2264<\/em>\u03bd(<em>U<\/em>) for every&nbsp;<em>U<\/em>&nbsp;\u2208&nbsp;<strong>O<\/strong><em>X<\/em>. &nbsp;With that ordering,&nbsp;<strong>V<em>X<\/em><\/strong>&nbsp;is a pointed dcpo, and Claire Jones proved that it is a continuous dcpo if&nbsp;<em>X<\/em>&nbsp;is a continuous dcpo. &nbsp;But&nbsp;<strong>V<em>X<\/em><\/strong>&nbsp;is not a complete lattice, and not a bc-domain, except in some specific cases: see [4] for some of the best results we know in this area, still today. &nbsp;(There were some others, including some that I am the author of, but this is not the topic I want to develop.)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Pretty recently, Mike Mislove proved that&nbsp;<strong>V<\/strong><em>P<\/em>&nbsp;is a continuous complete lattice in case&nbsp;<em>P<\/em>&nbsp;is a chain, in its Scott topology [5], with applications to a domain-theoretic approach to Skorokhod\u2019s theorem. &nbsp;Proposition A (or Corollary B) allows us to give a simple proof of this fact, and actually the following slight generalization.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Theorem.<\/strong>&nbsp;For every nested space&nbsp;<em>X<\/em>, the space&nbsp;<strong>V<\/strong><em>X<\/em>&nbsp;of continuous valuations on&nbsp;<em>X<\/em>&nbsp;is a continuous, complete lattice.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Proof.<\/em>\u00a0We observe that any Scott-continuous map from\u00a0<strong>O<\/strong><em>X<\/em>\u00a0to\u00a0<strong><span style=\"text-decoration: underline;\">R<\/span><\/strong><sub>+<\/sub>\u00a0is modular. Hence\u00a0<strong>V<\/strong><em>X<\/em>\u00a0can be equated with the dcpo of all the strict Scott-continuous maps from\u00a0<strong>O<\/strong><em>X<\/em>\u00a0to\u00a0<strong><span style=\"text-decoration: underline;\">R<\/span><\/strong><sub>+<\/sub>. Now we notice that\u00a0<strong>O<\/strong><em>X<\/em>\u00a0is a chain, hence is a continuous poset by Proposition A; it is also a complete lattice. \u00a0(Or: it is a chain and a pointed dcpo, hence a completely distributive complete lattice by Corollary B; in particular, a continuous, complete lattice again.) \u00a0Now, both\u00a0<strong>O<\/strong><em>X<\/em>\u00a0and\u00a0<strong><span style=\"text-decoration: underline;\">R<\/span><\/strong><sub>+<\/sub>\u00a0are objects in the Cartesian-closed category of continuous complete lattices (Exercise 5.7.17 in the <a href=\"https:\/\/www.cambridge.org\/gb\/knowledge\/isbn\/item7069109\/Non-Hausdorff%20Topology%20and%20Domain%20Theory\/?site_locale=en_GB\">book<\/a>), so the dcpo [<strong>O<\/strong><em>X<\/em>\u00a0\u2192\u00a0<strong><span style=\"text-decoration: underline;\">R<\/span><\/strong><sub>+<\/sub>] is also a continuous complete lattice. The subdcpo of those maps that are strict is closed under arbitrary suprema, and downwards-closed, and from that it is easy to deduce that it is also a continuous complete lattice. \u2610<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Roland Edwin Larson.&nbsp;&nbsp;<a href=\"https:\/\/msp.org\/pjm\/1969\/31-2\/pjm-v31-n2-p19-p.pdf\">Minimal T<sub>0<\/sub>&nbsp;spaces and minimal T<sub><em>D<\/em><\/sub>&nbsp;spaces<\/a>. Pacific Journal of Mathematics 31(2), 1969, pages 451-458.<\/li>\n\n\n\n<li>Edward Szpilrajn. &nbsp;<a href=\"https:\/\/eudml.org\/doc\/212499\">Sur l\u2019extension de l\u2019ordre partiel<\/a>. &nbsp;Fundamenta Mathematicae 16, pages 386\u2013389, 1930.<\/li>\n\n\n\n<li>Charles Edward Aull and Wolfgang Joseph Thron.&nbsp;<a href=\"https:\/\/core.ac.uk\/download\/pdf\/82702431.pdf\">Separation axioms between T<sub>0<\/sub>&nbsp;and T<sub>1<\/sub><\/a>. Indagationes Mathematicae 23, pages 26-37, 1962.<\/li>\n\n\n\n<li>Achim Jung and Regina Tix. <a href=\"https:\/\/www.cs.bham.ac.uk\/~axj\/pub\/papers\/Jung-Tix-1998-The-troublesome-probabilistic-powerdomain.pdf\">The Troublesome Probabilistic Powerdomain<\/a>. In A. Edalat, A. Jung, K. Keimel, and M. Kwiatkowska, editors,&nbsp;Proceedings of the Third Workshop on Computation and Approximation, volume 13 of&nbsp;Electronic Notes in Theoretical Computer Science. Elsevier Science Publishers B.V., 1998. 23 pages.<\/li>\n\n\n\n<li>Michael W. Mislove. Domains and stochastic processes. Theoretical Computer Science, 807, pages 284\u2013297, 2020. Available on&nbsp;<a href=\"https:\/\/arxiv.org\/abs\/1807.00884\">arXiv<\/a>.<\/li>\n<\/ol>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/wp-content\/uploads\/2016\/08\/jgl-2011.png\" alt=\"jgl-2011\" class=\"wp-image-993\" width=\"60\" height=\"83\"\/><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">\u2014 <a href=\"https:\/\/www.lsv.ens-paris-saclay.fr\/~goubault\/?l=en\">Jean Goubault-Larrecq<\/a> (August 23rd, 2020)<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A&nbsp;nested space&nbsp;is any topological space in which the lattice of open sets is totally ordered. &nbsp;My purpose this month is to show that this concept, which perhaps looks strange at first: I will start with a nifty, and perhaps surprising, &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=2702\">Continue reading <span class=\"meta-nav\">&rarr;<\/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-2702","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/2702","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=2702"}],"version-history":[{"count":26,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/2702\/revisions"}],"predecessor-version":[{"id":5898,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/2702\/revisions\/5898"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2702"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}