I finished my last post by saying that there were relationships between filter spaces and equilogical spaces.\u00a0 This was shown and elaborated upon by Reinhold Heckmann [1].\u00a0 As he notes, this had in fact been mentioned by Martin Hyland [2].\u00a0 Reinhold Heckmann gives a complete account of the matter, but goes through assemblies over algebraic complete lattices and modest sets, and I would like to give a simpler account here, if that is possible.<\/p>\n
From equilogical spaces to filter spaces<\/strong><\/p>\n Let (X<\/em>, \u2261) be an equilogical space.\u00a0 Since X<\/em> is a topological space, there is a natural notion of convergence on X<\/em>, let me write it \u2192. Instead of forming the quotient X<\/em>\/\u2261 in Top<\/strong>, one can instead for it in the category of filter spaces Filt<\/strong>.\u00a0 This way, we get a filter space from any equilogical space.\u00a0 That was quick!\u00a0 And we can check that this even provides a functor from Equ<\/strong> to Filt<\/strong>.<\/p>\n Before we try to go the other way around, and build equilogical spaces from filter spaces, let me describe this quotient a bit more explicitly.<\/p>\n Filt<\/strong> is not only Cartesian-closed, but also (complete and) cocomplete.\u00a0 The coproducts are built in the obvious way, and the coequalizers can be thought of as quotients, just like in Top<\/strong>. We build them as follows.\u00a0 Let q<\/em> : X <\/em>\u2192 X<\/em>\/\u2261 be the map that sends every point x<\/em> in X<\/em> to its equivalence class [x<\/em>].\u00a0 Given any filter F<\/em> of subsets of the quotient set X<\/em>\/\u2261, we can form the smallest filter of subsets of X<\/em> that contains all the subsets A<\/em> of X<\/em> whose direct image [A<\/em>]={[x<\/em>] |\u00a0x in A<\/em>} is in F<\/em>.\u00a0 Because q<\/em> is surjective, this is also the largest filter F’<\/em> of subsets of X<\/em> such that q<\/em>[F’<\/em>] is included in F<\/em>.\u00a0 (Exercise!)\u00a0 This filter F’<\/em> is the inverse image<\/em> filter of F<\/em> by q<\/em>.\u00a0 Now say that F<\/em> converges to [x<\/em>] in X<\/em>\/\u2261 if and only if the inverse image filter of F<\/em> by q<\/em> converges to some point equivalent to x<\/em> in X<\/em>.<\/p>\n This quotient has the usual universal property: q<\/em> itself is a continuous map between filter spaces, and every continuous map g<\/em> from X<\/em> to a filter space Y<\/em> such that g<\/em>(x)=g<\/em>(x’<\/em>) for all x<\/em>\u2261x’<\/em> factors uniquely through q<\/em>.<\/p>\n In the case that interests us, X<\/em> is a topological space, and convergence in X<\/em>\/\u2261 is described as follows: F<\/em> converges to [x<\/em>] in X<\/em>\/\u2261 if and only if there is a point x’<\/em>\u2261x<\/em> such that for every open neighborhood U<\/em> of x’<\/em> in X<\/em>, the direct image [U<\/em>] of U<\/em> is in F<\/em>.<\/p>\n In general, X<\/em>\/\u2261 will not be topological, even when X<\/em> is a topological space.\u00a0 To understand this, let me stress that quotients are computed differently in Filt<\/strong> and in Top<\/strong>.\u00a0 To make things clearer, let me write X<\/em>\/Filt<\/strong><\/sub>\u2261 for the quotient taken in Filt<\/strong>, and X<\/em>\/Top<\/strong><\/sub>\u2261 for the quotient taken in Top<\/strong>.\u00a0 While convergence in X<\/em>\/Filt<\/strong><\/sub>\u2261 is described above,\u00a0F<\/em> converges to [x<\/em>] in X<\/em>\/Top<\/strong><\/sub>\u2261 if and only if for every \u2261-saturated<\/em> open neighborhood U<\/em> of x<\/em>, [U<\/em>] is in F<\/em>.\u00a0 (We did not require U<\/em> to be \u2261-<\/em>saturated in X<\/em>\/Filt<\/strong><\/sub>\u2261.)<\/p>\n For an example, let N<\/strong>\u03c9<\/sub> be the dcpo of all natural numbers plus a fresh infinity element \u03c9 added, and take X<\/em> to be the topological coproduct N<\/strong>\u03c9<\/sub>+N<\/strong><\/strong>\u03c9<\/sub>.\u00a0 Let \u2261<\/em> equate the two copies of \u03c9, namely (0, \u03c9) and (1, \u03c9).\u00a0 The topological quotient\u00a0X<\/em>\/Top<\/strong><\/sub>\u2261 is the dcpo N<\/em>2<\/sub> of Figure 5.1, p.121.\u00a0 The filter F<\/em> of all neighborhoods of the equivalence class [(0, \u03c9)] (= [(1, \u03c9)]) converges to it in\u00a0X<\/em>\/Top<\/strong><\/sub>\u2261 (by definition), but it does not<\/em> converge to it in X<\/em>\/Filt<\/strong><\/sub>\u2261.\u00a0 The problem is that for every element x’<\/em> that is equivalent to (0, \u03c9) (let us say (0, \u03c9) itself, by symmetry), there is an open neighborhood U<\/em> of x’<\/em> in N<\/strong>\u03c9<\/sub>+N<\/strong><\/strong>\u03c9<\/sub>., say {42, 43, …, \u03c9} whose direct image [U<\/em>] is not in F<\/em>; in fact, you can check that [U<\/em>] has empty interior in N<\/em>2<\/sub>.\u00a0 I’ll let you extend the argument and show that F<\/em> has no limit whatsoever in X<\/em>\/Filt<\/strong><\/sub>\u2261.\u00a0 As a final exercise, show that whichever equilogical space (X<\/em>, \u2261<\/em>) is taken, convergence in X<\/em>\/Filt<\/strong><\/sub>\u2261 implies convergence in X<\/em>\/Top<\/strong><\/sub>\u2261; only the converse implication can fail (as in the example just given).<\/p>\n From filter spaces to equilogical spaces<\/strong><\/p>\n Conversely, let (Y<\/em>, \u2192<\/em>) be a filter space.\u00a0 Let me show you how one can build an equilogical space from it.\u00a0 (Something will go slightly wrong in the process, let me see whether you can spot it.)\u00a0 As mentioned earlier<\/a>, it is equivalent to find an algebraic complete lattice with a PER on it.<\/p>\n For the complete lattice, the natural choice is Filt<\/em>0<\/sub>(Y<\/em>), the poset of all filters on Y<\/em> under inclusion.\u00a0 This is very much related to the poset Filt<\/em>(Y<\/em>) we considered in Filters, part I<\/a>, except we also include the trivial filter now.\u00a0 R. Heckmann writes \u03a6(Y<\/em>) for this complete lattice.<\/p>\n This is a complete lattice indeed.\u00a0 Any intersection of filters is again a filter, and suprema of a family of filters is given by the filter generated by their union.\u00a0 As a special case, directed suprema are just unions, since a directed union of filters is already a filter.<\/p>\n We now check that Filt<\/em>0<\/sub>(Y<\/em>) is algebraic.\u00a0 For any subset\u00a0A<\/em> of Y<\/em>, and generalizing the notation (x<\/em>) we used in Filters, part II<\/a>, write (A<\/em>) for the filter of all supersets of A<\/em>. \u00a0(A<\/em>) is a finite element of Filt<\/em>0<\/sub>(Y<\/em>): if a directed union of filters contains the filter (A<\/em>), that is if A<\/em> is an element of the union of the filters, then A<\/em> is in one of them, which must therefore contain (A<\/em>).\u00a0 And every filter F<\/em> is the supremum of the filters (A<\/em>) when\u00a0A ranges over the elements of F<\/em>, and this family is directed since an upper bound of (A<\/em>) and (B<\/em>) with A<\/em>, B<\/em> in F<\/em> is given by (A<\/em> \u2229 B<\/em>).<\/p>\n Finally, we define a partial equivalence relation\u00a0\u2243<\/em> on Filt<\/em>0<\/sub>(Y<\/em>).\u00a0 Typically, we would like two filters to be equivalent if and only if they converge to the same limit in Y<\/em>, and those elements in dom \u2243<\/em> should be those that converge to exactly one limit.\u00a0 This runs into the problem that there may just not be any such filter.\u00a0 For example, in Z<\/strong>\u2014<\/sup>, the poset of non-positive integers with the Scott topology, all filters that converge to a point converge to all points below it, hence no filter converges to a unique point.<\/p>\n Instead, we shall consider so-called focused<\/em> filters.\u00a0 (I just coined the term).\u00a0 Say that a point y<\/em> in Y<\/em> is a focus point of a filter F<\/em> if and only if F<\/em> converges to y<\/em> and for every element A<\/em> of F<\/em>, y<\/em> is in A<\/em>.\u00a0 A filter with a focus point is called focused<\/em>.\u00a0 This makes sure we have an ample supply of focused filters: all the principal ultrafilters (y<\/em>) are focused, and y<\/em> is a focus point.\u00a0 (I’ll give further justification for focusing below.)<\/p>\n Following Heckmann, we now declare that F \u2243 <\/em>G<\/em>, for two filters F<\/em> and G<\/em>, if and only if both F<\/em> and G<\/em> are focused, and have a common focus point.<\/p>\n That is it, we now have an algebraic complete lattice Filt<\/em>0<\/sub>(Y<\/em>) with a PER \u2243<\/em>.\u00a0 We have already seen<\/a> that this was an equivalent way of describing an equilogical space.\u00a0 Explicitly, this is the topological space Foc<\/em>(Y<\/em>) of all focused filters of subsets of Y<\/em>, with the topology induced from the Scott topology on Filt<\/em>0<\/sub>(Y<\/em>), and with the equivalence relation \u2261 defined as meaning “have the same focus point”.\u00a0 I’ll also write Foc<\/em>(Y<\/em>) for the resulting equilogical space.<\/p>\n The topology on Foc<\/em>(Y<\/em>) is very simple, albeit a bit surprising.\u00a0 Since the sets \u2191(A<\/em>) form a base for the Scott topology on Filt<\/em>0<\/sub>(Y<\/em>), their intersections with Foc<\/em>(Y<\/em>) form a base of the latter.\u00a0 These are the sets \u2610A<\/em> of all focused filters F<\/em> that contain A<\/em>\u2014where A<\/em> is an arbitrary subset of Y<\/em>.<\/p>\n The whole construction is functorial, too.\u00a0 On morphisms f<\/em>: X<\/em> \u2192