let B is a basis for a topology on X. we have to show that the topology spanned by B is equal to the intersection of all topologies on x theed confecins B.
let B is a basis for a topology on X. we have to show that the topology spanned by B is equal to the intersection of all topologies on x theed confecins B.
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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![Led
let B is a basis for
a
topology
on X
we have to show that
the topology spanned by
B is equal to the intersection of call topologies
on x that confecins B
Lel {Tx} be the family of topologies
that contain B.
Consider the intersection Ta. It is clear
afd
that this intersection Contains & cand is a telology.
are shared,
Since for any two elements that
intersections and
unions must be shared,
Let U E TB = U = U Bi, where Bi EB
; E1
[: Bi's are basis elements
of BJ
and B; E O Ta
аєв
[":0₂ Ta Contains B]
аєв
·•. U = U ³³₁ € A Ta [Ta is a
аев
je]
Тв с пота
1-7
afß
доровду](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F59e64be3-c4a1-45fd-a5e6-8fcac7fe7d99%2Fcb60804a-a2b2-4118-bfb0-e7172cedb131%2Fephl10p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Led
let B is a basis for
a
topology
on X
we have to show that
the topology spanned by
B is equal to the intersection of call topologies
on x that confecins B
Lel {Tx} be the family of topologies
that contain B.
Consider the intersection Ta. It is clear
afd
that this intersection Contains & cand is a telology.
are shared,
Since for any two elements that
intersections and
unions must be shared,
Let U E TB = U = U Bi, where Bi EB
; E1
[: Bi's are basis elements
of BJ
and B; E O Ta
аєв
[":0₂ Ta Contains B]
аєв
·•. U = U ³³₁ € A Ta [Ta is a
аев
je]
Тв с пота
1-7
afß
доровду
Transcribed Image Text:Now
ule
Prove
seive that
п.та с ПВ
ded
Let U be an
Then U must be either in B or be
or intersection of elements of A.
a union
arbitrary element of n Ta.
aff
there in
Since
аса
acata is the largest tepo rosy containing
every topology contraining and by the definition
topology, this must be exactly the
of a
elements
of B and cenions and intersections
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Let is a basis for a topology on x.
we have to shown that the topology spanned by is equal to the intersection of all topologies on X that contains
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