1617-257/Homework Assignment 2 Solutions

DBN: Classes: 2016-17: 1617-257 / Navigation Welcome to Math 257 Edits to the Math 257 web pages no longer count for the purpose of good deed points. # Week of... Notes and Links 1 Sep 12 About This Class; Day 1 Handout (pdf, html); Monday, Wednesday, Friday, Day 2 Handout (pdf, html); First Week Notes. 2 Sep 19 Monday, Wednesday, Tutorial 2 Handout, Friday, Second week notes, HW1, HW1 Solutions. 3 Sep 26 Monday, Wednesday, Tutorial 3 Handout, Friday, Third week notes, Class Photo, HW2, HW2 Solutions. 4 Oct 3 Monday, Wednesday, Friday, 4th week notes, HW3, HW3 Solutions. 5 Oct 10 Monday is Thanksgiving, no class; Wednesday, Friday, 5th week notes, HW4, HW4 Solutions. 6 Oct 17 Monday, Wednesday, Friday, 6th week notes, HW5, HW5 Solutions. 7 Oct 24 Monday, Wednesday, Friday, 7th week notes 8 Oct 31 Monday; Term test 1; Wednesday, HW6, HW6 pdf, HW6 Solutions, Friday, 8th week notes. 9 Nov 7 Monday is last day to switch to MAT 237; Monday-Tuesday is UofT Fall Break; Wednesday, HW7, HW7 Solutions, Friday, 9th week notes. 10 Nov 14 Monday, Wednesday, HW8, HW8 pdf, HW8 Solutions, Friday, Lecture recordings, 10th week notes. 11 Nov 21 Monday, Wednesday, HW9, HW9 pdf, HW9 Solutions, Friday, Lecture recordings, 11th week notes. 12 Nov 28 Monday, Wednesday, HW10, HW10 Solutions, makeup class on Thursday at GB 120 at 5PM, no class and no DBN office hours Friday! 12th week notes. 13 Dec 5 Monday, Wednesday, 13th week notes Semester ends on Wednesday - no class Friday. B Dec 12,19,26 No classes: other classes' finals, winter break. 14 Jan 2 Class resumes Friday at RS211, no tutorials or office hours this week, Friday, Friday notes. 15 Jan 9 Monday, Wednesday, Friday, Weekly notes 16 Jan 16 Monday, Term test 2; Wednesday, HW11, HW11 inline pdf,HW11 solutions, Friday, Weekly notes 17 Jan 23 Monday; Hour 44 Handout (pdf, html); Wednesday, HW12, HW12 inline pdf, HW12 Solutions, Friday, Weekly notes 18 Jan 30 Monday, Wednesday, HW13, HW13 inline pdf, HW13 Solutions,Friday, Weekly Notes 19 Feb 6 Monday, Wednesday, HW14, HW14 inline pdf, Friday, Weekly Notes 20 Feb 13 Monday, Wednesday, HW15, HW15 inline pdf, HW15 solutions, Friday,Weekly Notes, UofT examination table posted on Friday. R Feb 20 Reading week - no classes; Tuesday is the last day to drop this class. 21 Feb 27 Monday, Wednesday, HW16, HW16 inline pdf, Friday, Weekly Notes 22 Mar 6 Monday, Wednesday, Friday, Weekly Notes 23 Mar 13 Monday, Term test 3 on Tuesday at 5-7PM; Wednesday, HW17, HW17 inline pdf, HW17 Solutions, Friday, Weekly Notes,Orientation_notes 24 Mar 20 Monday, Wednesday, HW18, HW18 inline pdf, Friday 25 Mar 27 No Monday class, Wednesday, Thursday (makeup for Monday), HW19, HW19 inline pdf, Friday, Weekly Notes 26 Apr 3 Monday, Wednesday; Semester ends on Wednesday - no tutorials Wednesday and Thursday, no class Friday. F Apr 10-28 The Final Exam on Thursday April 20 (and some office hours sessions before). Register of Good Deeds Riddle Repository Add your name / see who's in! $\int _{M}d\omega =\int _{\partial M}\omega$

Doing

Solve problems 1ab, 2, 3, 4ab in section 4, but submit only the underlined problems/parts. In addition, solve the following problems, though submit only your solutions of problems A and B:

Problem A. Let $(X,d)$ be a metric space. Prove that the metric itself, regarded as a function $d\colon X\times X\to {\mathbb {R} }$, is continuous.

Problem B. Let $A$ be a subset of a metric space $(X,d)$. Show that the distance function to $A$, defined by $d(x,A):=\inf _{y\in A}d(x,y)$, is a continuous function and that $d(x,A)=0$ iff $x\in {\bar {A}}$.

Problem C. Prove the "Lebesgue number lemma": If ${\mathcal {U}}=\{U_{\alpha }\}$ is an open cover of a compact space $(X,d)$, then there exists an $\epsilon >0$ (called "the Lebesgue number of ${\mathcal {U}}$, such that every open ball of radius $\epsilon$ in $X$ is contained in one of the $U_{\alpha }$'s.

Problem D. The Cantor set $C$ is the set formed from the closed unit interval $[0,1]$ by removing its open middle third $({\frac {1}{3}},{\frac {2}{3}})$, then removing the open middle thirds of the remaining two pieces (namely then removing $({\frac {1}{9}},{\frac {2}{9}})$ and $({\frac {7}{9}},{\frac {8}{9}})$), then removing the open middle thirds of the remaining 4 pieces, and so on. Prove that $C$ is uncountable, compact and totally disconnected (the last property means "the only non-empty connected subsets of $C$ are single points").

Student Solutions

Student 1