1617-257/Homework Assignment 2 - Revision history

Drorbn: /* Doing */

2016-11-09T17:26:00Z

Doing

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	<u>'''Problem B.'''</u> 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>.		<u>'''Problem B.'''</u> 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 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").		'''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").

Drorbn at 13:48, 2 November 2016

2016-11-02T13:48:54Z

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	{{1617-257/Navigation}}		{{1617-257/Navigation}}
	==Reading==		==Reading==
	Read, reread and rereread your notes to this point, and make sure that you really, really really, really really really understand everything in them. Do the same every week! Also, read, reread and rereread 3-4 of Munkres' book to the same standard of understanding. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, preread sections 5 and 6, just to get a feel for the future.		Read, reread and rereread your notes to this point, and make sure that you really, really really, really really really understand everything in them. Do the same every week! Also, read, reread and rereread sections 3-4 of Munkres' book to the same standard of understanding. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, preread sections 5 and 6, just to get a feel for the future.

	==Doing==		==Doing==

Drorbn at 16:09, 4 October 2016

2016-10-04T16:09:49Z

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	This assignment is due in class on <span style="color: blue;">Friday October 7 by 2:10PM</span>.		This assignment is due in class on <span style="color: blue;">Friday October 7 by 2:10PM</span>.

			===<span style="color: red;">Important</span>===
			<span style="color: red;">Please write on your assignment the day of the tutorial when you'd like to pick it up once it is marked (Wednesday or Thursday).</span>

Drorbn at 13:15, 1 October 2016

2016-10-01T13:15:40Z

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	{{In Preparation}}
	==Reading==		==Reading==
	Read, reread and rereread your notes to this point, and make sure that you really, really really, really really really understand everything in them. Do the same every week! Also, read, reread and rereread 3-4 of Munkres' book to the same standard of understanding. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, preread sections 5 and 6, just to get a feel for the future.		Read, reread and rereread your notes to this point, and make sure that you really, really really, really really really understand everything in them. Do the same every week! Also, read, reread and rereread 3-4 of Munkres' book to the same standard of understanding. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, preread sections 5 and 6, just to get a feel for the future.
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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").

			'''Also,''' Add your name to the [[1617-257/Class Photo\|Class Photo]] page!

	==Submission==		==Submission==

Drorbn: /* Doing */

2016-09-30T15:00:44Z

Doing

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	'''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 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 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").

	==Submission==		==Submission==

Drorbn at 14:55, 30 September 2016

2016-09-30T14:55:48Z

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	==Doing==		==Doing==
	'''Solve''' problems 1<u>a</u>bc ~~in section 1~~, ~~problems 2~~, <u>6</u> in section 2 ~~and~~ ~~problems~~ ~~4ab<u>c</u>,~~ ~~<u>6<~~/~~u>,~~ 7, ~~9abcd<u>efgh</u>~~ in ~~section~~ 3, ~~but~~ submit only ~~the~~ ~~underlined~~ problems~~/parts.~~		'''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:

			<u>'''Problem A.'''</u> 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.

			<u>'''Problem B.'''</u> 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 connected subsets of <math>C</math> are single points").

			==Submission==
	Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.		Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

Drorbn: Created page with "{{1617-257/Navigation}} {{In Preparation}} ==Reading== Read, reread and rereread your notes to this point, and make sure that you really, really really, really really really u..."

2016-09-30T14:01:44Z

Created page with "{{1617-257/Navigation}} {{In Preparation}} ==Reading== Read, reread and rereread your notes to this point, and make sure that you really, really really, really really really u..."

New page

{{1617-257/Navigation}}
{{In Preparation}}
==Reading==
Read, reread and rereread your notes to this point, and make sure that you really, really really, really really really understand everything in them. Do the same every week! Also, read, reread and rereread 3-4 of Munkres' book to the same standard of understanding. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, preread sections 5 and 6, just to get a feel for the future.

==Doing==
'''Solve''' problems 1abc in section 1, problems 2, 6 in section 2 and problems 4abc, 6, 7, 9abcdefgh in section 3, but submit only the underlined problems/parts.

Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

This assignment is due in class on Friday October 7 by 2:10PM.