{"id":12,"date":"2012-10-04T10:54:40","date_gmt":"2012-10-04T08:54:40","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=12"},"modified":"2023-09-23T14:13:22","modified_gmt":"2023-09-23T12:13:22","slug":"errata","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=12","title":{"rendered":"Errata"},"content":{"rendered":"

It is always embarrassing to realize that one’s mind has slipped…<\/p>\n

The important bloopers first:<\/h2>\n
    \n
  1. p.120, Definition 5.1.1 (the way-below relation): “for every directed family\u00a0zi<\/sub><\/em> that has a a least upper bound z<\/em> above y<\/em>, there is an i<\/em> in I<\/em> such that y<\/em> \u2264 zi<\/sub><\/em> already.”… \u00a0oops! such that x<\/em><\/strong>\u00a0\u2264\u00a0zi\u00a0<\/sub><\/em>already, not y<\/em>\u00a0\u2264\u00a0zi<\/sub><\/em>, as the subsequent explanation (p.121, top) states (found by\u00a0keith_dr_uk, May 14, 2013).<\/li>\n
  2. p.41, Exercise 3.5.8 and p.158, Exercise 5.4.17.\u00a0 Exercise 5.4.17 is false: the notion of continuous convergence used in Section 5.4 does not specialize to what is called continuous convergence in Exercise 3.5.8.\u00a0 (Consequence of a remark made by Barth Shiki, July 16, 2013.)
    \nCall weak<\/em> continuous convergence the notion of convergence used in Exercise 3.5.8.\u00a0 Continuous convergence of sequences implies<\/em> weak continuous convergence.\u00a0 The converse implication, which is asked for in Exercise 5.4.17, is unknown to me, but seems dubious.
    \nMy preferred fix is to modify Exercise 3.5.8 so as to give the right definition of continuous convergence there: (fn<\/sub><\/em>)n<\/sub><\/em> \u2208 N<\/sub> converges continuously to f<\/em> if and only if for every point x<\/em> and every sequence (xn<\/sub><\/em>)n<\/sub><\/em> \u2208 N<\/sub> of points of X<\/em> that converges to x<\/em>, for every \u03b5 > 0, there are two indices m<\/em>0<\/sub><\/em>, n0<\/sub><\/em><\/em> \u2208 N such that for all m \u2265 m<\/em>0<\/sub><\/em> and n \u2265 n<\/em>0<\/sub><\/em>, d<\/em> (f<\/em>(x<\/em>), fm<\/sub><\/em>(xn<\/sub><\/em>)) < \u03b5.\u00a0 Now show that if (fn<\/sub><\/em>)n<\/sub><\/em> \u2208 N<\/sub> converges continuously to f<\/em>, then f<\/em> is continuous (as suggested by Barth Shiki), and show (a) through (d), as before.
    \nOne should also modify Exercise 5.4.17, which should now read: Show that the notion of continuous convergence defined in Definition 5.4.15 coincides with the notion of Exercise 3.5.18 for sequences (fn<\/sub><\/em>)n<\/sub><\/em> \u2208 N<\/sub> of continuous maps from a metric space X<\/em> to a metric space Y<\/em>.<\/li>\n
  3. p.48, Lemma 4.1.10: “A family B<\/em> of subsets of a topological space X<\/em> is a base …”: should read “A family B<\/em> of open<\/strong> subsets of a topological space X<\/em> is a base …”.\u00a0 Otherwise, e.g., the family of all subsets of X<\/em> would be a base… (found, Dec 24, 2013).<\/li>\n
  4. p.121, Exercise 5.1.3: “Show that, in N<\/em>2<\/sub>, x<\/em> \u226a y<\/em> iff x<\/em> \u2264 y<\/em> and x<\/em>, y<\/em> \u2260 \u03c9” is wrong. You should instead show that the relation \u226a is empty<\/em>: x<\/em> \u226a y<\/em> for no<\/em> pair of elements x<\/em>, y<\/em>. In any case the conclusion that no element is way-below \u03c9 remains. (Found by Weng Kin Ho, Sep 04, 2014.)<\/li>\n
  5. p.156, Exercise 5.4.12. \u00a0You should ignore the last part of the exercise, which asks you to show that\u00a0Q<\/strong> and the Sorgenfrey line are not consonant. \u00a0That is true, but much more involved than the simplistic argument I am proposing. \u00a0As far as Q<\/strong> is concerned, see A. Bouziad, Borel measures in consonant spaces<\/em><\/a>, Topology and its Applications 70 (1996):125\u2014138; C. Costantini and S. Watson, On the dissonance of some metrizable spaces<\/em><\/a>,\u00a0Topology and its Applications 84 (1998):259\u2014268. \u00a0As far as the Sorgenfrey line is concerned, see B. Alleche and J. Calbrix,\u00a0On the coincidence of the upper Kuratowski topology with the cocompact topology<\/a>,<\/em>\u00a0Topology and its Applications 93(3):207\u2014218, 1999. \u00a0(found, Jul 10, 2015.) \u00a0See also this page<\/a> for the Sorgenfrey line (added, April 22, 2022.)<\/li>\n
  6. p.474, Exercise 9.7.53: the coefficient ring should be C<\/strong> throughout, not Z<\/strong>.\u00a0 Everything works with Z<\/strong> or C<\/strong> indifferently, except for Hilbert’s Nullstellensatz, which requires the coefficient ring to be an algebraically closed field.\u00a0 This is the case for C<\/strong>, but certainly not for Z<\/strong>.\u00a0 In particular, the final sentence should read “Using Kaplansky’s Theorem, show that, up to homeomorphism, the spectrum of Z<\/strong>[X<\/em>1<\/sub>, …, Xm<\/sub><\/em>] is the sobrification of C<\/strong>m<\/sup><\/em> in its Zariski topology.” (found, Nov 17, 2014.)<\/li>\n
  7. p.216, Exercise 6.3.10 and p.268, Exercise 7.2.4 and p.285, Exercise 7.3.12: the Sorgenfrey quasi-metric was defined with its arguments swapped.\u00a0 In other words, what I defined here is the opposite of the Sorgenfrey quasi-metric.\u00a0 The actual definition is: dl<\/sub>(r<\/em>,s<\/em>)=s<\/em>\u2014r<\/em> if s<\/em>\u2265r<\/em>, +\u221e otherwise.\u00a0 This is important if you want to retrieve the topology of the Sorgenfrey line in Exercise 6.3.10.
    \nThis has no consequence in the remaining places where the Sorgenfrey quasi-metric is mentioned, except in Exercise 7.3.12: you should replace x<\/em> by \u2014x<\/em>, essentially.\u00a0 That is, the required isomorphism should send (x<\/em>,r<\/em>) to (x<\/em>,\u2014x<\/em>\u2014r<\/em>), not to (\u2014x<\/em>,x<\/em>\u2014r<\/em>), you should show that the least upper bound of (xi<\/sub><\/em>, ri<\/sub><\/em>) is (supi<\/sub><\/em> xi<\/sub><\/em>, infi<\/sub><\/em> ri<\/sub><\/em>), and that (x<\/em>,r<\/em>) \u226a (y<\/em>,s<\/em>) iff x<\/em> < y<\/em> and y<\/em>\u2014x<\/em> < r<\/em>\u2014s<\/em>. (found, Mar 30, 2016.)<\/li>\n
  8. p.339, Exercise 7.7.20.\u00a0 The exercise is wrong.\u00a0 Indeed, the continuous open images of complete metric spaces are all first-countable (see Exercise 6.3.14), but there are spaces with a continuous model that are not first-countable.\u00a0 An example is given by {0,1}I<\/sup><\/em> for an uncountable set I<\/em>, which is not-first countable (see Exercise 6.5.9) but is compact Hausdorff, hence has a model by Corollary 8.3.27.\u00a0 I don’t know of any reasonable condition on Y<\/em> that would make the result true.\u00a0 (found by Ng Kok Min, Feb 26, 2017.)<\/li>\n
  9. p.387, Lemma 8.4.12.\u00a0 This only works provided Y<\/em> is T0<\/sub>.\u00a0 In other words, the equalizer of two continuous maps from a sober space X <\/em>to a\u00a0T0<\/sub> space\u00a0Y<\/em> is sober.\u00a0 If Y<\/em> is not required to be T0<\/sub>, then one can get absolutely any subspace A<\/em> of X<\/em> as an equalizer (let f<\/em> map the elements of A<\/em> to 1 and all others to 2, let g<\/em> map the elements of A<\/em> to 1 and all others to 3, where {1,2,3} is given the indiscrete topology).\u00a0 The proof is unchanged, except that at line 5, \u2264 is of course the specialization ordering of Y<\/em>, not X<\/em>, and that is an ordering because Y<\/em> is T0<\/sub>. (found by Zhenchao Lyu and Xiaodong Jia, Feb 08, 2019.)<\/li>\n
  10. p.463, Theorem 9.7.12, “the set \u2191x<\/em> \u2229 \u2191y<\/em>\u00a0of common upper bounds of x<\/em> and\u00a0y<\/em>” should read “the set \u2193x<\/em> \u2229 \u2193y<\/em>\u00a0of common lower bounds of x<\/em> and\u00a0y<\/em>“. \u00a0The same error occurs at line 5 of the second paragraph of the proof (found by\u00a0\u6c88\u51b2 [Shen Chong], May 24, 2019).<\/li>\n
  11. p.384, Lemma 8.4.5 and Theorem 8.4.6 are wrong. \u00a0The mistake is in line 3 of the proof of Lemma 8.4.5: yes,\u00a0S<\/strong> preserves coequalizers, but that means that\u00a0S<\/strong>(q<\/em>\u2261<\/sub>) is a coequalizer in the category\u00a0Sob<\/strong> of sober spaces, not<\/em> the category\u00a0Top<\/strong> of topological spaces… and coequalizers in\u00a0Sob<\/strong> need not be surjective. \u00a0In fact every<\/em> topological space at all can be obtained as a topological quotient of a sober space, and even of a Hausdorff space, by a result of M. Shimrat<\/a> of 1956.\u00a0 In general, coequalizers and colimits in\u00a0Sob<\/strong> are just what I said at the beginning of Section 8.4.1: sobrifications of the corresponding limits taken in\u00a0Top<\/strong>. \u00a0(found by Marcus Tressl, June 19, 2019)<\/li>\n
  12. p.85, Exercise 4.7.14. \u00a0It is asked to prove that “for every subset\u00a0A<\/em> of\u00a0X<\/em>,\u00a0A<\/em> is the set of all limits of sequences (xi<\/sub><\/em>)i<\/sub><\/em> \u2208 <\/sub>N<\/sub><\/strong>“. \u00a0That should read\u00a0“for every subset\u00a0A<\/em> of\u00a0X<\/em>, cl(A<\/em>) is the set of all limits of sequences (xi<\/sub><\/em>)i<\/sub><\/em> \u2208 <\/sub>N<\/sub><\/strong>\u00a0of points of A<\/em>“. \u00a0(found, December 05, 2019)<\/li>\n
  13. p.136, Proposition 5.1.60 (Scott’s formula), Corollary 5.1.61, Proposition 5.1.67, Proposition 5.1.69. \u00a0It is assumed that\u00a0Y<\/em> is a bdcpo for those to apply, but that is not enough. \u00a0You either need\u00a0Y<\/em> to be a dcpo, or\u00a0Y<\/em> to be a bdcpo and\u00a0B<\/em> to be cofinal in\u00a0X<\/em> (namely, every element of\u00a0X<\/em> is below some element of\u00a0B<\/em>; that happens if\u00a0B<\/em>=X<\/em>, notably). \u00a0The mistake lies in the first lines of the proof of Proposition 5.1.60: “… since {f<\/em>(b<\/em>) |\u00a0b<\/em> \u2208\u00a0B<\/em>,\u00a0b<\/em> \u226a\u00a0x<\/em>}\u00a0is directed, using the fact that f<\/em> is monotonic; and it has an upper bound, namely\u00a0f<\/em>(x<\/em>). \u00a0So it has a least upper bound, since Y<\/em> is a bdcpo.” \u00a0If\u00a0Y<\/em> is a dcpo, there is no problem: the family \u00a0{f<\/em>(b<\/em>) |\u00a0b<\/em> \u2208\u00a0B<\/em>,\u00a0b<\/em> \u226a\u00a0x<\/em>}\u00a0is directed, so it has a least upper bound. \u00a0If\u00a0Y<\/em> is a mere bdcpo,\u00a0f<\/em>(x<\/em>) is meaningless, since\u00a0f<\/em> only applies to elements of\u00a0B<\/em>. \u00a0But, if\u00a0B<\/em> is cofinal in\u00a0X<\/em>, then\u00a0x<\/em>\u2264b’<\/em> for some\u00a0b’<\/em> in\u00a0B<\/em>, and then\u00a0f<\/em>(b’<\/em>) is an upper bound of the family, which then has a least upper bound. \u00a0(found, February 7th, 2022.)<\/li>\n<\/ol>\n

    The less important ones second:<\/h2>\n
      \n
    1. p.25, Proposition 3.25, proof, second paragraph, second line: \u00a0“F<\/em>=\u22c2i in I<\/sub><\/em> Fi<\/sub><\/em>” should read “F<\/em>=\u22c3i in I<\/sub><\/em> Fi<\/sub><\/em>” (found by anon1, May 22, 2013).<\/li>\n
    2. p.396, Shmuely’s Theorem 8.4.29: I have been told it should be credited to George N. Raney (mentioned by A. Jung, March 22, 2013).\u00a0 Raney’s celebrated paper (1952) is https:\/\/www.jstor.org\/stable\/2032165<\/a>, but maybe this is not the right one: I’ve been unable to find it there yet.<\/li>\n
    3. p.478, Reference to Nachbin 1965, was reprinted by Robert E. Krieger Publishing Co., not “Robert E. Kreiger Publishing Co.” (May 13, 2013).<\/li>\n
    4. p.5, Union Axiom, definition of “\u222a x<\/em> = \u222ay<\/em> \u2208 x<\/em><\/sub> y<\/em>” is missing a condition on z<\/em>: this should be “\u222a x<\/em> = \u222ay<\/em> \u2208 x<\/em><\/sub> y<\/em> = {z<\/em> | \u2203 y<\/em> \u22c5 m<\/em> (y<\/em>) and z<\/em> \u2208 y<\/em> and y<\/em> \u2208 x<\/em>} is a set; i.e., m<\/em> ({z<\/em> | \u2203 y<\/em> \u22c5 m<\/em> (y<\/em>) and z<\/em> \u2208 y<\/em> and y<\/em> \u2208 x<\/em>})” (found by Barth Shiki, July 13, 2013).<\/li>\n
    5. p.20, Exercise 3.1.5, the hint is missing a power p<\/em> to the 1-\u03bb denominator: this should be |bi<\/sub><\/em>|p<\/sup><\/em> \/ (1-\u03bb)p<\/sup><\/em>, not |bi<\/sub><\/em>|p<\/sup><\/em> \/ (1-\u03bb) (found by Barth Shiki, July 16, 2013).<\/li>\n
    6. p.30, line 14: “d<\/em> (z<\/em>\u2192<\/sup>, y<\/em>\u2192<\/sup>) \u2264 2na<\/em>\/2k<\/sup><\/em>“: should be d<\/em> (z<\/em>\u2192<\/sup>, y<\/em>\u2192<\/sup>) \u2264 n<\/em> (2a<\/em>\/2k<\/sup><\/em>)p<\/sup><\/em>.\u00a0 Now also correct line 11 so that it say “let k<\/em> be such that n<\/em> (2a<\/em>\/2k<\/sup><\/em>)p<\/sup><\/em> < \u03b5p<\/sup><\/em>“, and the proof can proceed to the desired conclusion (found by Barth Shiki, July 16, 2013).<\/li>\n
    7. p.42, proof of Proposition 3.5.10, line 6: this implicitly uses the inequality d’<\/em> (f<\/em>(x’<\/em>), f<\/em>(x<\/em>)) \u2264 \u03b5\/3, and this was neither proved nor assumed (found by Barth Shiki, July 16, 2013).\u00a0 Add the following argument before the sentence starting with “Using the triangular inequality and the axiom of symmetry”: “From the inequality d’<\/em> (fn<\/sub><\/em><\/em>(x<\/em>), f<\/em>n<\/sub><\/em>(x’<\/em>)) \u2264 \u03b5\/3, we easily deduce d’<\/em> (f<\/em>(x<\/em>), f<\/em>(x’<\/em>)) \u2264 \u03b5\/3, by taking limits over n<\/em> \u2265 nx<\/sub><\/em><\/em><\/em>, with x<\/em> and x<\/em>‘ fixed; explicitly, for every \u03b5’ > 0, we have d’<\/em> (f<\/em>(x<\/em>), fn<\/sub><\/em><\/em>(x<\/em>)) < \u03b5’\/2 and d’<\/em> (f<\/em>(x’<\/em>), fn<\/sub><\/em><\/em>(x’<\/em>)) < \u03b5’\/2 for n<\/em> \u2265 nx<\/sub><\/em><\/em><\/em> large enough, so that the triangular inequality and the axiom of symmetry yield d’<\/em> (f<\/em>(x<\/em>), f<\/em>(x’<\/em>)) \u2264 d’<\/em> (f<\/em>(x<\/em>), fn<\/sub><\/em><\/em>(x<\/em>)) + d’<\/em> (fn<\/sub><\/em><\/em>(x<\/em>), f<\/em>n<\/sub><\/em>(x’<\/em>)) + d’<\/em> (fn<\/sub><\/em><\/em>(x’<\/em>), f<\/em>(x’<\/em>)) \u2264 \u03b5\/3 + \u03b5’; since \u03b5’ is arbitrary, d’<\/em> (f<\/em>(x<\/em>), f<\/em>(x’<\/em>)) \u2264 \u03b5\/3.”<\/li>\n
    8. p.43, proof of Theorem 3.5.12, l.10, there is a finite subset Ei<\/sub><\/em><\/em> such that is Kxi<\/sub><\/sub><\/em><\/em>, not K<\/em>, is included in the union of open balls centered at the points y<\/em> of Ei<\/sub><\/em><\/em> (found by Barth Shiki, July 16, 2013).<\/li>\n
    9. p.44, l.10, there is an argument missing at the end of this paragraph.\u00a0 We have shown that fn<\/sub><\/em><\/em> converged to f<\/em> uniformly, but we must also show that f<\/em> is in (calligraphic) K<\/em>.\u00a0 This follows from the unused (until now) assumption 3 (found by Barth Shiki, July 16, 2013).<\/li>\n
    10. p.61, right before Exercise 4.2.24, remove sentence “If so, then a subset U<\/em> of X<\/em> would be Scott-open iff it is upward closed and every chain having a least upper bound in U<\/em> meets U<\/em>.”\u00a0 Not only is it hard to see why it contributes to the argument, but this claim is actually true… see Exercise 4.2.26 (found by Barth Shiki, July 19, 2013).<\/li>\n
    11. p.65, remark after Proposition 4.3.9.\u00a0 More generally, every function from a T1<\/sub><\/em><\/em> space to an arbitrary space is monotonic (mentioned by Barth Shiki, July 19, 2013).<\/li>\n
    12. p.68, proof of Proposition 4.4.7, l.3, “such that Ui1<\/sub><\/sub><\/em>, …,\u00a0Uin<\/sub><\/sub><\/em> \u2286 U<\/em>” should be “such that Ui1<\/sub><\/sub><\/em>, …,\u00a0Uin<\/sub><\/sub><\/em> \u2286 Ui<\/sub><\/em>” (found by Barth Shiki, July 19, 2013).<\/li>\n
    13. p.71, proof of Proposition 4.4.17, lines 10-11, replace “F’<\/em> is compact, so extract a finite subcover (Vx<\/sub><\/em>)x<\/sub><\/em> \u2208 E<\/sub> of F’<\/em>” by “F<\/em> is compact, so extract a finite subcover (Ux<\/sub><\/em>)x<\/sub><\/em> \u2208 E<\/sub> of F<\/em>” (found by Barth Shiki, July 19, 2013).<\/li>\n
    14. p.312, Fact 7.5.23, there are extraneous ‘Y<\/strong>‘ subscripts to the categories YCQMet<\/strong> and YCQMetu<\/sub><\/strong>; they should be ignored (found, April 9, 2016).<\/li>\n
    15. p.34, before Theorem 3.4.5, and p.209, Definition 6.2.2: c<\/em>-Lipschitz maps are defined in the expected way, as those maps f<\/em> such that d<\/em>(f<\/em>(x<\/em>),f<\/em>(y<\/em>)) \u2264 c<\/em>.d<\/em>(x<\/em>,y<\/em>).\u00a0 This is however meaningless when c<\/em>=0 and d<\/em>(x<\/em>,y<\/em>)=+\u221e.\u00a0 This is repaired by taking the convention that 0.+\u221e=+\u221e.\u00a0 That convention is important: with that convention, f<\/em> is c<\/em>-Lipschitz if and only if B<\/strong>c<\/sup><\/em>(f<\/em>) is monotonic, and f<\/em> is c<\/em>-Lipschitz Yoneda-continuous if and only if B<\/strong>c<\/sup><\/em>(f<\/em>) is Scott-continuous (Proposition 7.4.38), but the equivalence would fail with any other convention (found, Sept. 14, 2016).<\/li>\n
    16. p.293, Lemma 7.4.17.\u00a0 The proof needs to be fixed (the Lemma holds).\u00a0 The problem stems from the fact that the family (ri<\/sub><\/em>)i \u2208 I<\/sub><\/em> is implicitly assumed to be filtered, which it need not be.\u00a0 This is repaired by noticing that the family\u00a0(ri<\/sub><\/em>+si<\/sub><\/em>)i \u2208 I<\/sub><\/em> is filtered.\u00a0 Here is the amended proof. Let ((xi<\/sub><\/em>, ri<\/sub><\/em>), si<\/sub><\/em>)i \u2208 I, \u2291<\/sub><\/em> be a Cauchy-weighted net in B<\/strong>(X<\/em>, d<\/em>).\u00a0 If i<\/em> \u2291 j<\/em> then d<\/em>+<\/sup> ((xi<\/sub><\/em>, ri<\/sub><\/em>), (xj<\/sub><\/em>, rj<\/sub><\/em>)) \u2264 si<\/sub><\/em>\u2013sj<\/sub><\/em>, and this implies both si<\/sub><\/em>\u2265sj<\/sub><\/em>, and d<\/em> (xi<\/sub><\/em>,xj<\/sub><\/em>) \u2264 ri<\/sub><\/em>+si<\/sub><\/em>\u2013(rj<\/sub><\/em>+sj<\/sub><\/em>); in particular, ri<\/sub><\/em>+si<\/sub><\/em>\u2265rj<\/sub><\/em>+sj<\/sub><\/em>.\u00a0 It follows that\u00a0(si<\/sub><\/em>)i \u2208 I<\/sub><\/em> and\u00a0(ri<\/sub><\/em>+si<\/sub><\/em>)i \u2208 I<\/sub><\/em> are filtered families. Let r<\/em>=infi \u2208 I<\/sub><\/em> (ri<\/sub><\/em>+si<\/sub><\/em>). Then (xi<\/sub><\/em>, ri<\/sub><\/em>+si<\/sub><\/em>\u2013r<\/em>)i \u2208 I, \u2291<\/sub><\/em> is Cauchy-weighted in X, d<\/em>.\u00a0 Let x<\/em> be the d<\/em>-limit of (xi<\/sub><\/em>)i \u2208 I, \u2291<\/sub><\/em>.\u00a0 By Lemma 7.4.9, d<\/em>(x<\/em>, y<\/em>)=supi \u2208 I<\/sub><\/em> (d<\/em>(xi<\/sub><\/em>, y<\/em>)\u2013ri<\/sub><\/em>\u2013si<\/sub><\/em>+r<\/em>)), a directed supremum, for every y<\/em> in X<\/em>.\u00a0 By a similar argument as in the book<\/a>, it follows that for every formal ball (y<\/em>, s<\/em>), d<\/em>+<\/sup> ((x<\/em>,\u00a0r<\/em>), (y<\/em>, s<\/em>)) = supi \u2208 I<\/sub><\/em> (d<\/em>+<\/sup> ((xi<\/sub><\/em>,\u00a0r<\/em>), (y<\/em>, s<\/em>)) \u2013 si<\/sub><\/em>), showing that (x<\/em>,\u00a0r<\/em>) is a d<\/em>+<\/sup>-limit of (xi<\/sub><\/em>, ri<\/sub><\/em>)i \u2208 I, \u2291<\/sub><\/em>.\u00a0 (found, Dec 1, 2016.)<\/li>\n
    17. p.267, Proposition 7.1.20: “Every limit of any net (xi<\/sub><\/em>)i \u2208 I, \u2291<\/sub><\/em> is a dop<\/sup><\/em>-limit” should read “Every limit of any net (xi<\/sub><\/em>)i \u2208 I, \u2291<\/sub><\/em> in X<\/em>, dsym<\/sup><\/em> is a dop<\/sup><\/em>-limit”, as the proof makes clear.\u00a0 (found, Jan 17, 2017.)<\/li>\n
    18. Section 7.5, p.311 and subsequent pages, the formal ball completion.\u00a0 One should give credit to Steve Vickers for that construction, see the reference below.\u00a0 I could say that I did not know of that when I wrote the book<\/a>, but that would not be true: Achim Jung had very kindly directed me to Steve’s work after I gave a talk on the subject at the Topology, Algebra, Categories and Logic conference in Marseilles in the summer of 2010… and I forgot about it.\u00a0 (found back, July 2016; inserted here, Jan 17, 2017.)\n