19. Let X be any nonempty set, and let B = {{x}: x € X}. Show that B is a base for a topology on X. Describe the open sets of this topology.
19. Let X be any nonempty set, and let B = {{x}: x € X}. Show that B is a base for a topology on X. Describe the open sets of this topology.
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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Question
Base of a topology: Please use certain theorems to solve this
![ex let X= R² and let Lx denote
the vertical line thru (x,0) in 1².
Let B=[LxXER].
Claim B is a base for
'
on
top-on X.
of Use Thm 2.26.
Each L₂ X=R², So B is a coll.
of subsets of X=IR³². Notice UB=IR³²
So holds in Thm 2.26. Show holds.
Assume B₁, B₂ EB and PE B, 0 B₂.
Then B₁-B₂. Choose C=B, EB.
So PEC=B, nBz. So B is a base
X=R². Call this
for
top-
X-A
145 U=U₁{²x: 0<x<i}
tu
let A= open unit disk
LET
A
= {(x,y): x² + y² <1}
int A=U open subsets of A/= Ø
Every nonempty
set is a
union of Belsen
-Claim [[0,2)/EIR² is in A'
←
meets S
Thm PES' every el. of B cont.
in a pt. other than
P
• Notice only LOEB Contains p
and
At Meets A. Lo meets A in a pt. other than
P.
"A² = {(x₁y): -1<x< 1}<
clA= AvA'A'.
top. T.
facts each LXET (each Lx is open)
LSUL, ET PEcls as every
el of B
cont.p meets & FrA = dl Ancl (x-A), = A'MIR' = A'](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc9787c00-cfcc-43da-9235-b2564f9dd12e%2F40732328-3f72-4d4f-aa4b-785dde0069a4%2F1vzeew_processed.jpeg&w=3840&q=75)
Transcribed Image Text:ex let X= R² and let Lx denote
the vertical line thru (x,0) in 1².
Let B=[LxXER].
Claim B is a base for
'
on
top-on X.
of Use Thm 2.26.
Each L₂ X=R², So B is a coll.
of subsets of X=IR³². Notice UB=IR³²
So holds in Thm 2.26. Show holds.
Assume B₁, B₂ EB and PE B, 0 B₂.
Then B₁-B₂. Choose C=B, EB.
So PEC=B, nBz. So B is a base
X=R². Call this
for
top-
X-A
145 U=U₁{²x: 0<x<i}
tu
let A= open unit disk
LET
A
= {(x,y): x² + y² <1}
int A=U open subsets of A/= Ø
Every nonempty
set is a
union of Belsen
-Claim [[0,2)/EIR² is in A'
←
meets S
Thm PES' every el. of B cont.
in a pt. other than
P
• Notice only LOEB Contains p
and
At Meets A. Lo meets A in a pt. other than
P.
"A² = {(x₁y): -1<x< 1}<
clA= AvA'A'.
top. T.
facts each LXET (each Lx is open)
LSUL, ET PEcls as every
el of B
cont.p meets & FrA = dl Ancl (x-A), = A'MIR' = A'
Transcribed Image Text:19. Let X be any nonempty set, and let B = {{x}: re X}. Show that B is a
base for a topology on X. Describe the open sets of this topology.
20 Sun
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