A property is a which remains intact under homeomorphism. We Let Let need to show 'Lindelof' topological property. f: x→Y Note that f: x→Y f-¹: y →x topological property Let By = {Vitie I of Yo be we are given with We shall Lindelof. both are Show be a Then Bo{ fi (U₁) Ying is Cover of homeo is a Continuous morphism. x Lindelof an an open Cover ~ (Y)=X. be open x is or, say Then, Lindelof, By has finite. infinite subcover countably By = { $* (U3) } je 1/ fr f.e Неп се, again infinite an x c f(x)=Y note that Y ус By = { Uj}se I' is an open sub cover of Y which is finite or countable By is so, U f (U5) SEI' =) Lindelof is UU; SEI' is Lindelof. U f(f(u) JEI' a topological property.
A property is a which remains intact under homeomorphism. We Let Let need to show 'Lindelof' topological property. f: x→Y Note that f: x→Y f-¹: y →x topological property Let By = {Vitie I of Yo be we are given with We shall Lindelof. both are Show be a Then Bo{ fi (U₁) Ying is Cover of homeo is a Continuous morphism. x Lindelof an an open Cover ~ (Y)=X. be open x is or, say Then, Lindelof, By has finite. infinite subcover countably By = { $* (U3) } je 1/ fr f.e Неп се, again infinite an x c f(x)=Y note that Y ус By = { Uj}se I' is an open sub cover of Y which is finite or countable By is so, U f (U5) SEI' =) Lindelof is UU; SEI' is Lindelof. U f(f(u) JEI' a topological property.
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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Transcribed Image Text:A
property is a
which remains intact under
homeomorphism.
We need to show 'Lindelof'
topological property
Let
fix → Y
fóx
Note that
Let
fix →Y
f-¹: y →x
we
topological property
We shall
Lindelof.
Let By = { uitie I
of Yo
be
both
are given with
are
be
an
show y to
Then Bo{ £" (Ui)ling is
= f-
Cover of
homeo
morphism.
Continuous
is a
x Lindelof
an
open Cover
f~1 (Y) = x.
be
open
3
as
say
Then
is
x is
or, countably infinite
2
F.e
Lindelof, Bx has finite
is again
B₂ = { ** (US)} SE I
ㅠ
Hence,
x
note that
infinite an
f(x)=Y
- ус
e
U f" (U5)
JEI'
an open sub cover of Y
finite or
countable
Lindelof is
subcover
U U;
JEI'
By is so,
U f (+ (15))
JEI'
B₂ = {US}5€I'
Y
y is Lindelof
which
topological
property.
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