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## ANALYSIS I

by: Cassidy Grimes

9

0

1

# ANALYSIS I MATH 554

Cassidy Grimes

GPA 3.51

R. Sharpley

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COURSE
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R. Sharpley
TYPE
Class Notes
PAGES
1
WORDS
KARMA
25 ?

## Popular in Mathematics (M)

This 1 page Class Notes was uploaded by Cassidy Grimes on Monday October 26, 2015. The Class Notes belongs to MATH 554 at University of South Carolina - Columbia taught by R. Sharpley in Fall. Since its upload, it has received 9 views. For similar materials see /class/229529/math-554-university-of-south-carolina-columbia in Mathematics (M) at University of South Carolina - Columbia.

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Date Created: 10/26/15
MATH 554 Handout 4 part B Example Each of the following is an example of a closed set a b c d Each closed interval 0 d is a closed subset of JR The set ood x E B x S d is a closed subset of IR Each singleton set 950 is a closed subset of B The Cantor set is a closed subset of IR To construct this set start with the closed interval 0 l and recursivley remove the open middle third of each of the remaining closed intervals At the n th stage we have a set E consisting of 2 closed intervals each of length Stage 0 Stage 1 0 Stage 2 07 ll This nite union of closed intervals is closed The Cantor set is the intersection of this decreasing or nested sequence of sets and so is also closed Later we will see that it has many other interesting properties We also showed in class that this set consists of real numbers whose ternary series expansions have coef cients which are drawn from 0 2 which allowed us to observe that the Cantor set is uncountable Finally we showed that the length of the corresponding set E at Stage 71 is 23 so the Cantor set has length equal limnnoo23 0

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