1. Prove that the set of all irrational numbers is dense in R. Hint: R is seperable so there exists a sequence of open sets that form a base. Use sequences, show that sequence is dense in R, therefore conclude set of irrationals is dense in R
1. Prove that the set of all irrational numbers is dense in R. Hint: R is seperable so there exists a sequence of open sets that form a base. Use sequences, show that sequence is dense in R, therefore conclude set of irrationals is dense in R
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(I included page from book that was said to follow idea as guide Theorem 5.10)
Thankyou, I dont know how prove using sequences
Transcribed Image Text:1. Prove that the set of all irrational numbers is dense in R.
Hint: R is seperable so there exists a sequence of open sets that form a base. Use sequences,
show that sequence is dense in R, therefore conclude set of irrationals is dense in R
Transcribed Image Text:union or
roIC oboce M
noiam
5.10 Theorem: A metric space is second-countable if and only if it is separable
Proof: Suppose first that X is a separable metric space. Let {x,}j=1 be a dense 5.12
sequence in X. Consider the family of open sets
bns ti
bos Yo
B
{B(x;1/n) : j > 1, n z 1}.
%3D
EXE
Let U be an open subset of X and let x e U. For some n 2 1, we have B(x;2/n) C
U. Choose j so that d(xjx) < 1/n. Then x e he
shows that B(x;1/n) C B(x;2/n) C U. Consequently B is a base of open sets. Since
B is countable, X is second-countable.
Conversely, suppose that X is second-countable. Let {U-1 be a sequence of
open sets in X that form a base. Let x, be any point in U,, n z 1. Then every
nonempty open subset of X contains a point of the sequence {x}, so that the sequence
is dense in X, and X is separable. O
€ B(x;1/n), and the triangle inequality
The following theorem is the trickiest part of the circle of ideas covered in this
section.
1gaiai doso bns Xax doso noi.Xlo isadue oldsin
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