64 N. H. Bingham <strong>and</strong> A. J. Ostaszewski(i) W ∩ Z Γ is dense in W ;(ii) ω(t) = 0 for all t ∈ W.In particular, if W = X, then X is <strong>topological</strong>.For a general W, as above,(i) if e X ∈ W <strong>and</strong> X is connected <strong>and</strong> Baire (under its norm topology), then X is a<strong>topological</strong> group,(ii) if X is separable, connected <strong>and</strong> <strong>topological</strong>ly complete (in its norm topology), thenX is a <strong>topological</strong> group.Proof. We first verify that (a)=⇒(b)=⇒(c)=⇒(a).Assume (a). Let ε > 0. Consider a nonempty U ⊆ W. Pick t ∈ U <strong>and</strong> suppose thatB η (t) ⊆ U with η < ε. By assumption, there is y η = y η (t) such that for some δ = δ(η) < ηwe haved R (tz, zy η ) ≤ η/2 for all ‖z‖ ≤ δ.Then in particular d R (t, y η ) ≤ η/2 < η, <strong>and</strong> also for all ‖z‖ ≤ δd R (y η z, zy η ) ≤ d R (y η z, tz) + d R (tz, zy η ) = d R (y η , t) + d R (tz, zy η ) ≤ η < ε.Thus y η ∈ B η (t) ∩ C(ε) ⊆ U ∩ C(ε). That is, (b) holds.Assume (b). Consider a non-empty U ⊆ W. Pick t ∈ U <strong>and</strong> suppose that B η (t) ⊆ U withη < ε/3. By assumption, there is y η = y η (t) ∈ B η (t) such that for some δ = δ(η) < η wehaved R (y η z, zy η ) ≤ η for all ‖z‖ ≤ δ.We prove that y η is a 3η-shifting point <strong>and</strong> so a ε-shifting point, i.e. thatIndeed, we haved R (y η z, y η ) ≤ 3η for all ‖z‖ ≤ δ.d R (y η z, y η ) ≤ d R (y η z, zy η ) + d R (zy η , y η ) = d R (y η z, zy η ) + d R (z, e)≤ 2η + δ < 3η < ε.Thus, y η ∈ B η (t) ∩ Ω(ε) ⊆ U ∩ Ω(ε). That is, (c) holds.Now suppose that (c) holds. Consider t ∈ W <strong>and</strong> ε > 0. Suppose that B η (t) ⊆ W withη < ε/2. By assumption, there is y η = y η (t) such that for some δ = δ(η) < η we haveHenced R (y η z, zy η ) ≤ η for all ‖z‖ ≤ δ.d R (tz, zy η ) ≤ d R (tz, y η z) + d R (y η z, zy η ) ≤ 2η < ε.So for y = y η , we have d R (tz, zy) < ε for all ‖z‖ ≤ δ. Thus (a) holds.Now that we have verified the equivalences, suppose that (a) holds.From (c), for t ∈ W <strong>and</strong> any ε > 0, we have ω(t) ≤ ω δ(ε) (t) ≤ ε. As ε > 0 was arbitrary,we have ω(t) = 0. Hence if W = X, then X = Z Γ <strong>and</strong> the group is <strong>topological</strong>. The othertwo conclusions follow from Th. 3.44 <strong>and</strong> 3.4˙5.
<strong>Normed</strong> <strong>groups</strong> 65Remark. In the penultimate step above with W = X, one can take ε > 0; then for0 < η ≤ ε we have d(t, y η (t)) ≤ η, <strong>and</strong> ω(y η ) < 3η ≤ 3ε, so the points {y η (t) : t ∈ X, 0 0, S(ε) = X, i.e. every point is an ε-shifting point for any ε > 0.In particular, X is a <strong>topological</strong> group.Proof. Suppose not. Then there is x <strong>and</strong> ε > 0 such that x is not ε-shifting, i.e. for eachn there is z n ∈ B 1/n (x) such thatd(x, xz n ) ≥ ε.Let η < ε/4. Since z n is null <strong>and</strong> B η (x) is open (so Baire) <strong>and</strong> non-meagre, by theassumed KBD there are t ∈ B η (x) <strong>and</strong> an infinite M t such that tz m ∈ B η (x) for m ∈ M t .So, since d(t, tz m ) < 2η, for any m ∈ M td R (x, xz m ) ≤ d R (x, t) + d R (t, tz m ) + d R (tz m , xz m )= 2d R (x, t) + d R (t, tz m ) < 4η < ε,a contradiction.Thus X = S(ε), for each ε > 0. By Th. 3. 49, X is a <strong>topological</strong> group.Theorem. 3.51 is a corollary of the Dense Oscillation Theorem (Th. 3.48) <strong>and</strong> indicatesa ‘Darboux-like’ pathology when the normed group is not <strong>topological</strong>.Theorem 3.51 (Pathology Theorem). If a normed group X is not a <strong>topological</strong> group,then there is an open set on which the oscillation function is uniformly bounded awayfrom 0.Proof. This follows from the continuity of ω at any point t where ω(t) > 0. This alsofollows from Th. 3.48, since for some ε > 0, the open set U := X\cl[Ω(ε)] is non-empty,<strong>and</strong> ω(t) ≥ ε for t ∈ U, as t /∈ Ω(ε).By way of a final clarification of our interest in ε-shifting points, we return to theliterature of ‘separate implies joint continuity’ <strong>and</strong> in particular to the key notion ofquasi-continuity, which we adapt here to a metric context (for further information seee.g. [Bou2]).
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N. H. BINGHAM and A. J. OSTASZEWSKI
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Normed groups 3ContentsContents . .
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1. IntroductionGroup-norms, which b
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Normed groups 3Topological complete
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Normed groups 5abelian group has se
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Normed groups 74 (Topological permu
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Normed groups 9The following result
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Normed groups 11Corollary 2.4. For
- Page 17 and 18: Normed groups 13More generally, for
- Page 19 and 20: Normed groups 15definitions, our pr
- Page 21 and 22: Normed groups 17so that fg is in th
- Page 23 and 24: Normed groups 19(iii) The ¯d H -to
- Page 25 and 26: Normed groups 21so‖αβ‖ ≤
- Page 27 and 28: Normed groups 23Remark. Note that,
- Page 29 and 30: Normed groups 25shows that [z n , y
- Page 31 and 32: Normed groups 27Denoting this commo
- Page 33 and 34: Normed groups 29Theorem 3.4 (Equiva
- Page 35 and 36: Normed groups 31argument as again p
- Page 37 and 38: Normed groups 33(ii) For α ∈ H u
- Page 39 and 40: Normed groups 35Definition. A group
- Page 41 and 42: Normed groups 37We now give an expl
- Page 43 and 44: Normed groups 39Theorem 3.19 (Abeli
- Page 45 and 46: Normed groups 412. Further recall t
- Page 47 and 48: Normed groups 43Theorem 3.22 (Lipsc
- Page 49 and 50: Normed groups 45Proof. Z γ = G (cf
- Page 51 and 52: Normed groups 47Theorem 3.30. Let G
- Page 53 and 54: Normed groups 49Remark. On the matt
- Page 55 and 56: Normed groups 51As for the conclusi
- Page 57 and 58: Normed groups 53By (C-adm), we may
- Page 59 and 60: Normed groups 55equipped with an in
- Page 61 and 62: Normed groups 57Proof. To apply Th.
- Page 63 and 64: Normed groups 59Definition. A point
- Page 65 and 66: Normed groups 61Proposition 3.46 (M
- Page 67: Normed groups 63Thus ω δ (s) ≤
- Page 71 and 72: Normed groups 67The result confirms
- Page 73 and 74: Normed groups 69Proof. By the Baire
- Page 75 and 76: Normed groups 715. Generic Dichotom
- Page 77 and 78: Normed groups 73Returning to the cr
- Page 79 and 80: Normed groups 75Examples. Here are
- Page 81 and 82: Normed groups 77cf. [Eng, 4.3.23].)
- Page 83 and 84: Normed groups 79Remarks. 1. See [Fo
- Page 85 and 86: Normed groups 81Theorem 6.1 (Catego
- Page 87 and 88: Normed groups 83is continuous at th
- Page 89 and 90: Normed groups 85compact. Evidently,
- Page 91 and 92: Normed groups 87j ∈ ω} which enu
- Page 93 and 94: Normed groups 89The result below ge
- Page 95 and 96: Normed groups 91left-shift, not in
- Page 97 and 98: Normed groups 93As a corollary of t
- Page 99 and 100: Normed groups 953. For X a normed g
- Page 101 and 102: Normed groups 97Proof. Note that‖
- Page 103 and 104: Normed groups 99Taking h(x) := ‖
- Page 105 and 106: Normed groups 1019. The Semigroup T
- Page 107 and 108: Normed groups 103Theorem 9.5 (Semig
- Page 109 and 110: Normed groups 105By the Category Em
- Page 111 and 112: Normed groups 107Proof. Say f is bo
- Page 113 and 114: Normed groups 109Thus G is locally
- Page 115 and 116: Normed groups 111Theorem 10.10 (Bar
- Page 117 and 118: Normed groups 113K-analyticity was
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Normed groups 115Theorem 11.6 (Disc
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Normed groups 117restricted to X\M
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Normed groups 119groups need not be
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Normed groups 121Proof. In the meas
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Normed groups 123Hence, as t i n
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Normed groups 125The corresponding
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Normed groups 127(t, x) ✛✻Φ T
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Normed groups 129Fix s. Since s is
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Normed groups 131Hence,‖x‖ −
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Normed groups 133converging to the
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Normed groups 135Definition. Let {
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Normed groups 137However, whilst th
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Normed groups 139embeddable, 14enab
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Normed groups 141Bibliography[AL]J.
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Normed groups 143Series 378, 2010.[
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Normed groups 145abelian groups, Ma
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Normed groups 147[Kak] S. Kakutani,
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Normed groups 149fields. I. Basic p
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Normed groups 151[So]R. M. Solovay,