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What can you say about a set ACR, if every subset of A is closed in R? Which of the following subsets of R are complete? Connected? Compact? A = Z, B = {x € R : a° – 3x³ < 4}, C = {x € R : sinx € Z}.

What can you say about a set ACR, if every subset of A is closed in R? Which of the following subsets of R are complete? Connected? Compact? A = Z, B = {x € R : a° – 3x³ < 4}, C = {x € R : sinx € Z}.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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3. What can you say about a set A C R, if every subset of A is closed in R? 4. Which of the following subsets of R are complete? Connected? Compact? A = Z, B = {x € R : x°. 3.x < 4}, C = {x €R: sin x € Z}. 5. Suppose that A, B CR are dense and open in R. Show that ANB is dense in R. 6. The distance between a point x E R and a nonempty set A CR is defined by d(x, A) = inf {\x – y| : y € A}. Show that d(x, A) = 0 if and only if x is an element of the closure A.
Transcribed Image Text:3. What can you say about a set A C R, if every subset of A is closed in R? 4. Which of the following subsets of R are complete? Connected? Compact? A = Z, B = {x € R : x°. 3.x < 4}, C = {x €R: sin x € Z}. 5. Suppose that A, B CR are dense and open in R. Show that ANB is dense in R. 6. The distance between a point x E R and a nonempty set A CR is defined by d(x, A) = inf {\x – y| : y € A}. Show that d(x, A) = 0 if and only if x is an element of the closure A.
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