Maggie: /* Student Solutions */

2016-11-25T20:50:44Z

Student Solutions

← Older revision		Revision as of 16:50, 25 November 2016
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	{{1617-257/Navigation}}		{{1617-257/Navigation}}

			==Doing==
⚫	==Student Solutions==
	'''Solve''' problems 1a<u>b</u>, 2, 3, 4a<u>b</u> 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:		'''Solve''' problems 1a<u>b</u>, 2, 3, 4a<u>b</u> 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:

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	'''Problem D.''' The ''Cantor set'' <math>C</math> is the set formed from the closed unit interval <math>[0,1]</math> by removing its open middle third <math>(\frac13,\frac23)</math>, then removing the open middle thirds of the remaining two pieces (namely then removing <math>(\frac19,\frac29)</math> and <math>(\frac79,\frac89)</math>), then removing the open middle thirds of the remaining 4 pieces, and so on. Prove that <math>C</math> is uncountable, compact and totally disconnected (the last property means "the only non-empty connected subsets of <math>C</math> are single points").		'''Problem D.''' The ''Cantor set'' <math>C</math> is the set formed from the closed unit interval <math>[0,1]</math> by removing its open middle third <math>(\frac13,\frac23)</math>, then removing the open middle thirds of the remaining two pieces (namely then removing <math>(\frac19,\frac29)</math> and <math>(\frac79,\frac89)</math>), then removing the open middle thirds of the remaining 4 pieces, and so on. Prove that <math>C</math> is uncountable, compact and totally disconnected (the last property means "the only non-empty connected subsets of <math>C</math> are single points").

		⚫	==Student Solutions==

	~~Student 1:~~ [[Media:1617-257_homework2_solutions.pdf\|~~HW2~~ ~~Sol~~]]		[[Media:1617-257_homework2_solutions.pdf\|Student 1]]

	Student 2:

Abhi.saim: Created page with "{{1617-257/Navigation}} ==Student Solutions== '''Solve''' problems 1ab, 2, 3, 4ab in section 4, but submit only the underlined problems/parts. In addition, solv..."

2016-10-30T00:00:45Z

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{{1617-257/Navigation}}

==Student Solutions==
'''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 <math>(X,d)</math> be a metric space. Prove that the metric itself, regarded as a function <math>d\colon X\times X\to{\mathbb R}</math>, is continuous.

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

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

'''Problem D.''' The ''Cantor set'' <math>C</math> is the set formed from the closed unit interval <math>[0,1]</math> by removing its open middle third <math>(\frac13,\frac23)</math>, then removing the open middle thirds of the remaining two pieces (namely then removing <math>(\frac19,\frac29)</math> and <math>(\frac79,\frac89)</math>), then removing the open middle thirds of the remaining 4 pieces, and so on. Prove that <math>C</math> is uncountable, compact and totally disconnected (the last property means "the only non-empty connected subsets of <math>C</math> are single points").

Student 1: [[Media:1617-257_homework2_solutions.pdf|HW2 Sol]]

Student 2:

1617-257/Homework Assignment 2 Solutions - Revision history

Maggie: /* Student Solutions */

Abhi.saim: Created page with "{{1617-257/Navigation}} ==Student Solutions== '''Solve''' problems 1ab, 2, 3, 4ab in section 4, but submit only the underlined problems/parts. In addition, solv..."