1617-257/Homework Assignment 2 Solutions: Difference between revisions
(Created page with "{{1617-257/Navigation}} ==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, solv...") |
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==Doing== |
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'''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"). |
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[[Media:1617-257_homework2_solutions.pdf|Student 1]] |
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Student 2: |
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Latest revision as of 16:50, 25 November 2016
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 be a metric space. Prove that the metric itself, regarded as a function , is continuous.
Problem B. Let be a subset of a metric space . Show that the distance function to , defined by , is a continuous function and that iff .
Problem C. Prove the "Lebesgue number lemma": If is an open cover of a compact space , then there exists an (called "the Lebesgue number of , such that every open ball of radius in is contained in one of the 's.
Problem D. The Cantor set is the set formed from the closed unit interval by removing its open middle third , then removing the open middle thirds of the remaining two pieces (namely then removing and ), then removing the open middle thirds of the remaining 4 pieces, and so on. Prove that is uncountable, compact and totally disconnected (the last property means "the only non-empty connected subsets of are single points").
![{\displaystyle [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d)
