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13 May 2016: If φ : (Sn,) (B,B) , show that we can find a mapΦ : X E such that p Φ = φ π and Φ(Sn1 0) =. If we define φ =Φ|Sn11 : Sn1 ... F , check that the map [φ] 7 [φ] descends to a well-definedhomomorphism : πn(B,B) πn1(F,).

16 Jun 2016: ben1 = inf(B) inf(A) = m1. Line I should be. nk1 = inf{b B : b > nk} inf{a A : a > mk} = mk1. ... The formula in Page 47, Line 1 should be. nk1 = inf{b B : b > nk} mk < mk1.

7 Oct 2016: Show that f : Q Q withf ( a. b) = b.

11 Nov 2016: u(0,x) =1. 1 x2. 1. 5. (Cauchy-Kovalevskaya Theorem for systems of ODEs) Suppose b > 0 and F : u0 (b,b)d Rd bereal analytic in a neighbourhood of ... u′(t) = F(t,u(t)), u(0) = u0 Rd,. on (a,a) for some a > 0 with u((a,a)) u0 (b,b)d.

23 Mar 2016: B = ( B ,. , B , e l ,. , e , ) and C = ( C 1 ,. , C , e ,. , e , ). T h e n i r a • 1,4,1,. ... Recall t h a t if B = ( B ,. , B, b l ,. , b,) is a E-algebra, then a value a of t y p e [¢Ix] interpreted at B I ,. , B

23 Oct 2016: φ(s,v) = α(s) r(n(s) cos v b(s) sin v). where n = n(s) and b = b(s) are the normal and binormal vectors of α. ... 10. Consider a surface of revolution parametrized by φ : (0, 2π) (a,b) R3, where.

27 Oct 2016: algebra. 1.2. Show that the following sets of subsets of R all generate the same σ-algebra:(a) {(a,b) : a < b}, (b) {(a,b] : a < b}, (c) {(,b] : b ... 1.8. Let B be a Borel subset of the interval [0, 1].

27 Oct 2016: Hint: Bolzano-Weierstrass.). 3. If A and B are subsets of Rn, let A B = {a b |a A,b B}. ... a) f(x) = sin x2 (b) f(x) = inf{|xn2| |n Z} (c) (sin x3)/(x 1).

29 Feb 2016: 8. By a selection function for a set a, we mean a function s: Pa Pa such that s(b) b for allb a, both inclusions being strict unless b is ... By considering congruence classes whose members are well-orderings, show that theHartogs ordinal of b is greater

23 Mar 2016: B = ( B ,. , B , e l ,. , e , ) and C = ( C 1 ,. , C , e ,. , e , ). T h e n i r a • 1,4,1,. ... Recall t h a t if B = ( B ,. , B, b l ,. , b,) is a E-algebra, then a value a of t y p e [¢Ix] interpreted at B I ,. , B