{"id":283,"date":"2013-11-22T14:48:13","date_gmt":"2013-11-22T13:48:13","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=283"},"modified":"2022-11-19T15:32:20","modified_gmt":"2022-11-19T14:32:20","slug":"filters-part-ii-filter-spaces","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=283","title":{"rendered":"Filters, part II: filter spaces"},"content":{"rendered":"<p>I said earlier that what convergence was the starting point of topology.\u00a0 Why not take this seriously and replace topological spaces by spaces that would be defined in terms of notions of convergence <em>directly<\/em>, instead of through opens?<\/p>\n<p>This works well, both with Moore-Smith convergence (in terms of nets) and with filters.\u00a0 One bonus is that we shall obtain Cartesian-closed categories, and this will be easy!\u00a0 Comparing with our exploration of exponential topologies (Section 5.4), of <em>C<\/em>-generated spaces (Section 5.6), of bc-domains (Section 5.7), this will be a relief.<\/p>\n<p>Oh, just to set it straight: there will be a snag with the net convergence approach.\u00a0 This will force us to use filters in the end.\u00a0 Since I still feel nets are more easily understandable, let&#8217;s start with them.<\/p>\n<p><strong>Net convergence spaces.<\/strong><\/p>\n<p>Let us implement our program of defining a replacement for topological spaces that would be based on convergence directly.<\/p>\n<p>Call a <em>net convergence space<\/em> any set <em>X<\/em> together with a relation <em>\u2192<\/em>, between nets (<em>x<sub>i<\/sub><\/em>)<em><sub>i<\/sub><\/em><sub> \u2208 <\/sub><em><sub>I, \u2291<\/sub><\/em> and points <em>x<\/em> in <em>X<\/em>, satisfying the following two axioms, due to Garrett Birkhoff [1, Theorem 7]:<\/p>\n<ul>\n<li>(4\u03b1) If <em>x<sub>i<\/sub><\/em>=<em>x<\/em> for every<em> i<\/em> in <em>I<\/em>, then (<em>x<sub>i<\/sub><\/em>)<em><sub>i<\/sub><\/em><sub> \u2208 <\/sub><em><sub>I, \u2291<\/sub><\/em> \u2192 <em>x<\/em><\/li>\n<li>(4\u03b2) If (<em>x<sub>i<\/sub><\/em>)<em><sub>i<\/sub><\/em><sub> \u2208 <\/sub><em><sub>I<\/sub><\/em><sub>, \u2291<\/sub> \u2192 <em>x<\/em>, then (<em>x<sub>\u03b1 (j)<\/sub><\/em>)<em><sub>j<\/sub><\/em><sub> \u2208 <\/sub><em><sub>J, \u2264<\/sub><\/em> \u2192 <em>x<\/em> for every subnet (<em>x<sub>\u03b1 (j)<\/sub><\/em>)<em><sub>j<\/sub><\/em><sub> \u2208 <\/sub><em><sub>J, \u2264<\/sub><\/em>.<\/li>\n<\/ul>\n<p>The phrase (<em>x<sub>i<\/sub><\/em>)<em><sub>i<\/sub><\/em><sub> \u2208 <\/sub><em><sub>I, \u2291<\/sub><\/em> \u2192 <em>x<\/em> would mean that the net (<em>x<sub>i<\/sub><\/em>)<em><sub>i<\/sub><\/em><sub> \u2208 <\/sub><em><sub>I, \u2291<\/sub><\/em> converges to x, of course.\u00a0 Axiom (4\u03b1) states that any constant net converges to the obvious value, while (4\u03b2) states that convergence and limits are preserved by taking subnets.<\/p>\n<p>Net convergence spaces form a category, whose morphisms are those maps that preserve convergence.\u00a0 Call such maps continuous: a continuous map is a map\u00a0<em>f<\/em> such that if (<em>x<sub>i<\/sub><\/em>)<em><sub>i<\/sub><\/em><sub> \u2208 <\/sub><em><sub>I<\/sub><\/em><sub>, \u2291<\/sub> \u2192 <em>x<\/em> then (<em>f<\/em>(<em>x<sub>i<\/sub><\/em>))<em><sub>i<\/sub><\/em><sub> \u2208 <\/sub><em><sub>I<\/sub><\/em><sub>, \u2291<\/sub> \u2192 <em>f<\/em>(<em>x<\/em>).\u00a0 (For those of you who are impatient, the announced snag lies somewhere in this very paragraph.)<\/p>\n<p>Products in this category are defined in the obvious way: a net of tuples converges if and only if it converges componentwise.<\/p>\n<p>Moreover, this category is Cartesian-closed: the exponential object [<em>X<\/em> \u2192 <em>Y<\/em>] is just the space of continuous maps from <em>X<\/em> to <em>Y<\/em> with the structure of <em>continuous convergence<\/em>&#8230; yes, continuous convergence of nets (Section 5.4.1), namely (<em>f<sub>i<\/sub><\/em>)<em><sub>i<\/sub><\/em><sub> \u2208 <\/sub><em><sub>I<\/sub><\/em><sub>, \u2291<\/sub> \u2192 <em>f<\/em> in [<em>X<\/em> \u2192 <em>Y<\/em>] if and only if for every convergence net (<em>x<sub>j<\/sub><\/em>)<em><sub>j<\/sub><\/em><sub> \u2208 <\/sub><em><sub>J, \u2264<\/sub><\/em> \u2192 <em>x, <\/em>(<em><em>f<sub>i<\/sub><\/em><\/em>(<em><em>x<sub>j<\/sub><\/em><\/em>))<em><em><sub>(i,j)<\/sub><\/em><\/em><sub> \u2208 <\/sub><em><em><sub>I \u00d7 J, <\/sub><\/em><\/em><sub>\u2291 \u00d7 \u2264<\/sub><em> \u2192 <em>f<\/em><\/em>(<em><em>x<\/em><\/em>) in <em>Y<\/em>.<\/p>\n<p>The convergence relation on [<em>X<\/em> \u2192 <em>Y<\/em>] is in effect defined as the least one that makes App continuous.\u00a0 That&#8217;s it.<\/p>\n<p>Much simpler than the situation in topological spaces, right?<\/p>\n<p>But there is a snag, as I said earlier.\u00a0 Have you spotted it?<\/p>\n<p>OK, here it is.<\/p>\n<p>The problem is that nets on a set <em>X<\/em> do not form a set, but a proper class.\u00a0 This is because there are at least as many nets (<em>x<sub>i<\/sub><\/em>)<em><sub>i<\/sub><\/em><sub> \u2208 <\/sub><em><sub>I<\/sub><\/em><sub>, \u2291<\/sub> on <em>X<\/em> as there are indexing sets <em>I<\/em>, and the class of all sets is not a set.\u00a0 In the VBG set theory that I&#8217;m using as foundations in the <a href=\"https:\/\/www.cambridge.org\/gb\/knowledge\/isbn\/item7069109\/Non-Hausdorff%20Topology%20and%20Domain%20Theory\/?site_locale=en_GB\">book<\/a>, you cannot then even define what a relation \u2192 would be!\u00a0 A binary relation R between two sets <em>A<\/em> and <em>B<\/em> is indeed a subset of <em>A<\/em> \u00d7 <em>B<\/em>.\u00a0 Here we would need to define \u2192 as a subset of the binary product of the class of nets with <em>X<\/em>.\u00a0 But VBG set theory has no provision for forming a product of a proper class with anything: it is just too large.<\/p>\n<p>There are logical ways of fixing this.\u00a0 Use <a title=\"Tarski-Grothendieck set theory\" href=\"https:\/\/en.wikipedia.org\/wiki\/Tarski%E2%80%93Grothendieck_set_theory\">Tarski-Grothendieck<\/a> set theory (TG), or <a title=\"Morse-Kelley set theory\" href=\"https:\/\/en.wikipedia.org\/wiki\/Morse%E2%80%93Kelley_set_theory\">Morse-Kelley<\/a> set theory (MK), for example.\u00a0 The latter was even used in topology.\u00a0 But I&#8217;d like to stick to a set theory that is as standard as it can be.\u00a0 VBG is a good compromise: as a conservative extension of standard Zermelo-Fraenkel with choice (ZFC) set theory, everything I say in VBG set theory can be recast in ZFC set theory, and VBG gives me the added comfort of being able to talk about classes explicitly.\u00a0 TG and MK are non-conservative extensions, meaning that some statements you can prove in TG or MK are not provable in ZFC.<\/p>\n<p><strong>Filter spaces.<\/strong><\/p>\n<p>All this is repaired by using filters. The class of filters on a set <em>is a set<\/em>, getting rid of the problem with nets.\u00a0 And of course, filters are enough to define convergence.\u00a0 We are no longer leaving the cosy realm of standard axioms for Mathematics.\u00a0 The only price we have to pay is to use filter convergence, which is more obscure than net convergence.<\/p>\n<p>OK, let&#8217;s redo what we&#8217;ve done above, this time in the language of filters.<\/p>\n<p>A <em>filter space<\/em> is a pair (<em>X<\/em>, <em>\u2192<\/em>) where X is a set and <em>\u2192<\/em> is a relation between <em>filters<\/em> <em>F<\/em> and points <em>x<\/em>, satisfying the following two axioms, analogous to the ones given above for nets:<\/p>\n<ul>\n<li>(<em>x<\/em>) <em>\u2192<\/em> <em>x<\/em>, where (<em>x<\/em>) is the filter of all subsets of <em>X<\/em> that contain <em>x<\/em> (this is called the <em>principal ultrafilter at<\/em> <em>x<\/em>)<\/li>\n<li>If <em>F<\/em> <em>\u2192<\/em> <em>x<\/em> and <em>F&#8217;<\/em> is a filter that contains <em>F<\/em>, then <em>F&#8217;<\/em> <em>\u2192<\/em> <em>x<\/em>.<\/li>\n<\/ul>\n<p>My reference is Hyland [2], who does wonders with them in the theory of computation.\u00a0 I don&#8217;t know who invented them.\u00a0 Variants of these (limit spaces, convergence spaces) already existed before, as one reckons by reading Hyland.<\/p>\n<p>These are the only axioms we are requiring.\u00a0 One could add extra axioms, such as: if <em>F<\/em> <em>\u2192<\/em> <em>x<\/em> and <em>F&#8217;<\/em> <em>\u2192<\/em> <em>x<\/em> then <em>F<\/em> \u2229 <em>F&#8217; <em>\u2192<\/em><\/em> <em>x<\/em>.\u00a0 This is an example of a property that is true in every topological space, but may fail in filter spaces.\u00a0 Incidentally, filter spaces with this extra property are called <em>convergence spaces<\/em>.<\/p>\n<p><strong>The category of filter spaces.<\/strong><\/p>\n<p>Filter spaces form a category, whose morphisms are the continuous maps: a map from a filter space <em>X<\/em> to a filter space <em>Y<\/em> is continuous if and only if whenever <em>F<\/em> <em>\u2192<\/em> <em>x<\/em> in <em>X<\/em>, then <em>f<\/em>[<em>F<\/em>] <em>\u2192<\/em> <em>f<\/em>(<em>x<\/em>) in <em>Y<\/em>.\u00a0 Recall that <em>f<\/em>[<em>F<\/em>] = {<em>B<\/em> \u2286 <em>Y<\/em> | <em>f<\/em><sup>\u20131<\/sup> (<em>B<\/em>) is in <em>F<\/em>} is the image filter of <em>F<\/em> by <em>f<\/em>.<\/p>\n<p>One of the nice properties of filter spaces is that they form a Cartesian closed category.\u00a0 This is as for net convergence spaces&#8230; but this is really working now.<\/p>\n<p>The binary product <em>X<\/em> \u00d7 <em>Y<\/em> is the usual Cartesian product, with convergence defined by <em>F<\/em> \u2192 (<em>x<\/em>, <em>y<\/em>) iff \u03c0<sub>1<\/sub>[<em>F<\/em>] \u2192 x and \u03c0<sub>2<\/sub>[<em>F<\/em>] \u2192 y.\u00a0 This is componentwise convergence, as expected.\u00a0 In particular, if <em>F<\/em><sub>1<\/sub> \u2192 <em>x<\/em> in <em>X<\/em> and <em>F<\/em><sub>2<\/sub> \u2192 <em>y<\/em> in <em>Y<\/em>, then <em>F<\/em><sub>1<\/sub> \u2297 <em>F<\/em><sub>2<\/sub> \u2192 (<em>x<\/em>, <em>y<\/em>), where <em>F<\/em><sub>1<\/sub> \u2297 <em>F<\/em><sub>2<\/sub> is the filter of all those sets that contain a rectangle <em>U<\/em> \u00d7 <em>V<\/em> with <em>U<\/em> in <em>F<\/em><sub>1<\/sub> and <em>V<\/em> in <em>F<\/em><sub>2<\/sub>: one recognizes the usual pattern defining the product topology.<\/p>\n<p>The exponential object [<em>X<\/em> \u2192 <em>Y<\/em>] is just the space of continuous maps from <em>X<\/em> to <em>Y<\/em> with the structure of <em>continuous convergence<\/em>&#8230; yes, the translation of continuous convergence of nets (Section 5.4.1) into the language of filters: a filter <em>F<\/em> of continuous maps from <em>X<\/em> to <em>Y<\/em> <em>converges continuously<\/em> to <em>f<\/em> if and only if, for every filter <em>G<\/em> that converges to a point <em>x<\/em> in <em>X<\/em>, <em>F<\/em>(<em>G<\/em>) converges to <em>f<\/em>(x) in <em>Y<\/em>.\u00a0 The filter <em>F<\/em>(<em>G<\/em>) is just App[<em>F<\/em> \u2297 <em>G<\/em>], where App is the usual application map, so the convergence relation on [<em>X<\/em> \u2192 <em>Y<\/em>] is in effect defined as the least one that makes App continuous.\u00a0 That&#8217;s all.<\/p>\n<p><strong>Relation to topological spaces.<\/strong><\/p>\n<p>Every topological space <em>X<\/em> can be converted to a filter space <em>Flt<\/em>(<em>X<\/em>), by letting \u2192 be the usual notion of convergence in <em>X<\/em> (&#8220;<em>F<\/em> \u2192 <em>x<\/em> iff <em>F<\/em> contains the neighborhood filter <em>N<sub>x<\/sub><\/em>&#8220;).\u00a0 Every continuous map <em>f\u00a0<\/em>:\u00a0<em>X<\/em>\u00a0\u2192 <em>Y<\/em> between topological spaces defines a\u00a0 map <em>Flt<\/em>(<em>f<\/em>) (=<em>f<\/em>) between filter spaces, which is continuous in the sense that it preserves the convergence relation.\u00a0 This makes <em>Flt<\/em> a functor.<\/p>\n<p>Conversely, any filter space <em>X<\/em>, as defined above, defines a topological space <em>Top<\/em>(<em>X<\/em>): we let its opens be those subsets <em>U<\/em> of <em>X<\/em> such that for every point <em>x<\/em> in <em>U<\/em>, for every filter\u00a0<em>F<\/em> such that <em>F<\/em> \u2192 <em>x<\/em>, <em>U<\/em> is in <em>F<\/em>.\u00a0 This also defines a functor <em>Top<\/em>.<\/p>\n<p><em>Top<\/em> and <em>Flt<\/em> are not inverses of each other: in fact there are many more filter spaces than topological spaces.\u00a0 But <em>Top<\/em> is left adjoint to <em>Flt<\/em>, so that one can say that <em>Top<\/em>(<em>X<\/em>) is the <em>free<\/em> topological space over the filter space <em>X<\/em>.<\/p>\n<p>One checks easily that the counit of the adjunction is the identity map: <em>Top<\/em>(<em>Flt<\/em>(<em>X<\/em>)) is just the topological space\u00a0<em>X<\/em>.\u00a0 The unit \u03b7:<em>X\u00a0\u2192\u00a0<em>Flt<\/em><\/em>(<em><em>Top<\/em><\/em>(<em><em>X<\/em><\/em>)), where <em>X<\/em> is a filter space, is more interesting.\u00a0 At the level of points, it is just again the identity map, but more filters converge in <em><em>Flt<\/em><\/em>(<em><em>Top<\/em><\/em>(<em><em>X<\/em><\/em>)) than in <em>X<\/em>.<\/p>\n<p>Call a filter space <em>topological<\/em> if and only if \u03b7 is iso, equivalently, the topological filter spaces are those of the form <em><em>Top<\/em><\/em>(<em><em>X<\/em><\/em>) for some topological space.\u00a0 This allows us to consider every topological space <em>X<\/em> as a topological filter space, equating <em>X<\/em> with <em><em>Top<\/em><\/em>(<em><em>X<\/em><\/em>).\u00a0 But there are more filter spaces than topological spaces.<\/p>\n<p>A quick argument is as follows.\u00a0 Let X be a non-core compact topological space, so that there is a topological space <em>Y<\/em> such that [<em>X<\/em> \u2192 <em>Y<\/em>] has no exponential topology.\u00a0 It is an easy exercise to show that the exponential object [<em><em><em>Flt<\/em><\/em><\/em>(<em>X<\/em>) \u2192 <em><em><em>Flt<\/em><\/em><\/em>(<em>Y<\/em>)] in the category of filter spaces cannot be topological.<\/p>\n<p><strong>An example: Scott-convergence.<\/strong><\/p>\n<p>Let us give another, more concrete example of non-topological filter spaces.\u00a0 These ones will be familiar to domain-theorists (or see Section 4.2).\u00a0 We start with the net convergence view.<\/p>\n<p>Let <em>X<\/em> be a dcpo.\u00a0 We can define <em>Scott-convergence<\/em> of nets (<em>x<sub>i<\/sub><\/em>)<em><sub>i<\/sub><\/em><sub> \u2208 <\/sub><em><sub>I<\/sub><\/em><sub>, \u2291<\/sub> in X by (<em>x<sub>i<\/sub><\/em>)<em><sub>i<\/sub><\/em><sub> \u2208 <\/sub><em><sub>I<\/sub><\/em><sub>, \u2291<\/sub><em> \u2192 x<\/em> if and only if (<em>x<sub>i<\/sub><\/em>)<em><sub>i<\/sub><\/em><sub> \u2208 <\/sub><em><sub>I<\/sub><\/em><sub>, \u2291<\/sub> is a monotone net, and <em>x <\/em>is below its sup (not <em>equal<\/em> to its sup: we really mean that <em>x<\/em> is a limit of\u00a0(<em>x<sub>i<\/sub><\/em>)<em><sub>i<\/sub><\/em><sub> in <\/sub><em><sub>I, \u2291<\/sub><\/em> in the Scott topology.)\u00a0 This obeys Birkhoff&#8217;s axioms (4\u03b1) and (4\u03b2).\u00a0 The topology defined by this notion of convergence, which is defined as usual as the collection of subsets such that if (<em>x<sub>i<\/sub><\/em>)<em><sub>i<\/sub><\/em><sub> \u2208 <\/sub><em><sub>I<\/sub><\/em><sub>, \u2291<\/sub><em> \u2192 x<\/em> \u2208 <em>U<\/em> then\u00a0<em>x<sub>i<\/sub><\/em> is in <em>U<\/em> for <em>i<\/em> large enough, is exactly the Scott topology.\u00a0 But topological convergence (for the Scott topology) is strictly larger than our original notion of convergence: there are nets that converge topologically, but not with our original definition of <em>\u2192<\/em>, simply because they are not <em>monotone<\/em> nets.\u00a0 For example, consider the sequence of intervals [1\u20131\/<em>n<\/em>, 1\u20131\/<em>n<\/em>] with <em>n<\/em> positive natural number in <strong>I<\/strong>R, Scott&#8217;s domain of closed intervals of the real line ordered by reverse inclusion (Ex. 4.2.28).\u00a0 These intervals can be equated with the reals 1\u20131\/<em>n<\/em>, which converge to 1 in the usual topology on R, hence to [1, 1] in the Scott topology.\u00a0 But the sequence of these intervals does not converge to anything with respect to <em>\u2192<\/em>, since it is not a monotone net in <strong>I<\/strong>R.<\/p>\n<p>Imitating this with filters, let&#8217;s say that <em>F \u2192 x<\/em> (<em>Scott-convergence<\/em> of filters) if and only if:<\/p>\n<ul>\n<li><em>F<\/em> is the convergence filter of some monotone net, and<\/li>\n<li><em>x<\/em> is below the sup of this monotone net.<\/li>\n<\/ul>\n<p style=\"text-align: left;\">Recall that the convergence filter of a net (<em>x<sub>i<\/sub><\/em>)<em><sub>i<\/sub><\/em><sub> \u2208 <\/sub><em><sub>I<\/sub><\/em><sub>, \u2291<\/sub> is the collection of supersets of sets of the form {<em>x<sub>i<\/sub><\/em> | <em>i<\/em> \u2265 <em>i<\/em><sub>0<\/sub>} with <em>i<\/em><sub>0<\/sub> in <em>I<\/em>.\u00a0 The second condition can be rewritten as: <em>x<\/em> is a lower bound of the\u00a0intersection \u22c2<em><sub>A <\/sub><\/em><sub>\u2208<\/sub><em><sub>F<\/sub><\/em>\u2191<em>A<\/em> of the upward closures of elements of <em>F<\/em>, which shows that it is independent of the chosen monotone net.<\/p>\n<p style=\"text-align: left;\">As above with nets, the topology defined by this notion of convergence is the Scott topology, so <em>Top<\/em>(<em>X<\/em>) really is <em>X<\/em> with its Scott topology.\u00a0 The notion of convergence in <em><em>Flt<\/em><\/em>(<em><em>Top<\/em><\/em>(<em><em>X<\/em><\/em>)) is larger than Scott-convergence.\u00a0 In <strong>I<\/strong>R, the convergence filter <em>F<\/em><sub>1\u2013<\/sub>\u00a0of the sequence of intervals [1\u20131\/<em>n<\/em>, 1\u20131\/<em>n<\/em>] converges to [1, 1] in the Scott topology, that is, in <em><em>Flt<\/em><\/em>(<em><em>Top<\/em><\/em>(<em><em>X<\/em><\/em>)); but it does not converge to anything in <em>X<\/em>, because <em>F<\/em><sub>1\u2013<\/sub> is not the convergence filter of any monotone net.<\/p>\n<p style=\"text-align: left;\"><strong>Topological filter spaces.<\/strong><\/p>\n<p>Recall that we could equate each topological space <em>X<\/em> with the filter space\u00a0<em>Flt<\/em>(<em>X<\/em>).\u00a0 Making this identification, the adjunction\u00a0<em>Top<\/em> \u22a3 <em>Flt<\/em> can be read by saying that the category of topological spaces is a reflective subcategory of the category of filter spaces.<\/p>\n<p>Can we characterize those filter spaces that are topological?\u00a0 Remember that every topological filter space is a convergence space, meaning that if <em>F<\/em> <em>\u2192<\/em> <em>x<\/em> and <em>F&#8217;<\/em> <em>\u2192<\/em> <em>x<\/em> then <em>F<\/em> \u2229 <em>F&#8217; <em>\u2192<\/em><\/em> <em>x<\/em>.\u00a0 (By the way, you can check that Scott-convergence of filters, as defined above, does not in general define a structure of a convergence space, giving another argument why Scott-convergence is not topological.)\u00a0 The same property holds even for arbitrary, not just binary, intersections.<br \/>\nIn other words, every topological filter space satisfies the much stronger axiom:<\/p>\n<ul>\n<li>if (<em>F<sub>i<\/sub><\/em>)<em><sub>i<\/sub><\/em><sub> \u2208 <\/sub><em><sub>I<\/sub><\/em> is any family of filters on <em>X<\/em> such that\u00a0<em>F<sub>i<\/sub><\/em> <em>\u2192<\/em> <em>x<\/em> for every <em>i<\/em> in <em>I<\/em>, then <em>F <\/em><em>\u2192<\/em> <em>x<\/em> where <em>F<\/em> is the intersection of the <em>F<sub>i<\/sub><\/em>s.<\/li>\n<\/ul>\n<p>Conversely, any filter space that satisfies the latter axiom is topological.\u00a0 Consider indeed the family of all the filters that converge to <em>x<\/em>, and take their intersection.\u00a0 It is natural to call this intersection filter <em>N<sub>x<\/sub><\/em>, as it will turn out to be the filter of open neighborhoods of <em>x<\/em>.\u00a0 The extra axiom above implies that <em>N<sub>x<\/sub><\/em> <em>\u2192<\/em> <em>x<\/em>, and then that a filter converges to <em>x<\/em> iff it contains <em>N<sub>x<\/sub><\/em>.\u00a0 From <em>N<sub>x<\/sub><\/em>, we retrieve a topology by declaring open any set <em>U<\/em> that is a neighborhood of each of its points, i.e., such that <em>U<\/em> is in <em>N<sub>x<\/sub><\/em> for every <em>x<\/em> in <em>U<\/em>: in particular, <em>N<sub>x<\/sub><\/em> is really the filter of neighborhoods of <em>x<\/em>: our filter space is indeed topological.<\/p>\n<p>Oops, there is mistake in there!\u00a0 We shall see in <a title=\"Filters, part III\" href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=319\">Filters, part III<\/a> where the error lies.\u00a0 Let us have some rest until then.<\/p>\n<p style=\"text-align: right;\">\u2014 <a href=\"https:\/\/www.lsv.ens-paris-saclay.fr\/~goubault\/?l=en\" rel=\"attachment wp-att-993\">Jean Goubault-Larrecq<\/a>\u00a0(November 22nd, 2013)<img loading=\"lazy\" decoding=\"async\" class=\"wp-image-993 alignright\" src=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/wp-content\/uploads\/2016\/08\/jgl-2011.png\" alt=\"jgl-2011\" width=\"32\" height=\"44\" \/><\/p>\n<ol>\n<li>Garrett Birkhoff. <em>Moore-Smith Convergence in General Topology<\/em>.\u00a0 The Annals of Mathematics, Second Series, Vol. 38, No. 1 (Jan., 1937), pp. 39-56.<\/li>\n<li>J. Martin E. Hyland. <em>Filter Spaces and Continuous Functionals<\/em>.\u00a0 Annals of Mathematical Logic, Vol. 16 (1979), pp. 101-143.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>I said earlier that what convergence was the starting point of topology.\u00a0 Why not take this seriously and replace topological spaces by spaces that would be defined in terms of notions of convergence directly, instead of through opens? This works &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=283\">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":"open","ping_status":"open","template":"","meta":{"_crdt_document":"","footnotes":""},"class_list":["post-283","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/283","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=283"}],"version-history":[{"count":14,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/283\/revisions"}],"predecessor-version":[{"id":5967,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/283\/revisions\/5967"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=283"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}