suppose, f: (₁R, Zu) → (IR, Te) be homeomorphism
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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![NOW
Suppose,
Suppose
IR, set of real numbers with usual topology Zu
Чи
Z₁ = {(9,6):
ась, яьназ
IR with finite complement topology Te
f homeomorphism
Since,
But, by
че
{UCR RIU finite subset of IR
{R}
CR
f: (IR, Tu) → (IR, Te) be homeomorphism.
[0,1]closed Set (closed interval) in (MR, Tu).
As, f: (¹2, Tu) - (IR, Te) i homeomorphism,
in (iR, Tc).
f ([0, 1]) is closed set
=
=) f 3
[0,1] infinite set
f 3 bijective.
(ie, cofinite topology).
Hence, + ([0,1])
f
=) f ([0,1]) i infinite set.
Closed set in (112,
defination of finite complement topology,
only finite sets can be closed set in (12. (e).
Hence, contradiction happens.
([0,1]) i infinite
SO (IR, Tu) and (R, Tc)
50
are not homeomorphic.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc52fbd7b-9adb-4e52-81b8-0bf956b4f8e7%2F3dddcbfe-c93d-4f83-b86e-c679b8af5bc5%2F4vbt8h5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:NOW
Suppose,
Suppose
IR, set of real numbers with usual topology Zu
Чи
Z₁ = {(9,6):
ась, яьназ
IR with finite complement topology Te
f homeomorphism
Since,
But, by
че
{UCR RIU finite subset of IR
{R}
CR
f: (IR, Tu) → (IR, Te) be homeomorphism.
[0,1]closed Set (closed interval) in (MR, Tu).
As, f: (¹2, Tu) - (IR, Te) i homeomorphism,
in (iR, Tc).
f ([0, 1]) is closed set
=
=) f 3
[0,1] infinite set
f 3 bijective.
(ie, cofinite topology).
Hence, + ([0,1])
f
=) f ([0,1]) i infinite set.
Closed set in (112,
defination of finite complement topology,
only finite sets can be closed set in (12. (e).
Hence, contradiction happens.
([0,1]) i infinite
SO (IR, Tu) and (R, Tc)
50
are not homeomorphic.
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