the integers. Find the of N. 2. Let X be the set of reals, and let T = {SCX:0€ X-S} U{X}. Show that T is a topology for X and find the closure of the interval A = (1, 2) and of the interval B = (-1,1). 3. Let X be the set of positive integers. For each n E X, let Sn = {k € X:
the integers. Find the of N. 2. Let X be the set of reals, and let T = {SCX:0€ X-S} U{X}. Show that T is a topology for X and find the closure of the interval A = (1, 2) and of the interval B = (-1,1). 3. Let X be the set of positive integers. For each n E X, let Sn = {k € X:
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Topology
Please follow guide in 2nd photo to solve problem
Transcribed Image Text:Def Let (XT) be a top. space. If SEX,
T-)tmit
then pех на
pt. of Sif
meets S
very
in
a
el. of T cont. p
P
pt. Other than
ex X=IR, T= {0, Ⓡ
Suppole SER and SØ.
Show cI(S)= IR.
Ple+ (pEX=R). The only el. of
is IR and
T cont.
Thm 2.2 let (X₁T) be a top. Space. If SEX
[PECIS) every el. of I cont. p
then
meets S
Thm 23 let (X, TJ be a top. space. If SCX,
Sis T-closed S'ES
then
ef→ Assume (SEX is T-closed) Show S'ES.
Notice S'ES'US = CISS
Assume (S'ES Show S=CIS
"C" SESUSECI (5)
"2" cls=SUSSUS = S
P
IR meets S. By Thm 2.2,
PECI(S). SoX=R)=cl S.
Show c1(S) ≤ X=IR.
CI S=SUS ER UR = IR
Then clS= R
Claim R is closed in (X,T)
ef Since IR SIR and R #ø,
we get cI (IR) = IR ₁
So R is closed.
Claim if SER
is a proper subset
of IR, then S
is NOT closed.
Transcribed Image Text:the integers. Find the
of N.
2. Let X be the set of reals, and let T = {SCX:0€ X-S} U{X}. Show
that T is a topology for X and find the closure of the interval A = (1, 2)
and of the interval B = (-1,1).
3. Let X be the set of positive integers. For each n E X, let Sn = {kE X:
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