Box Topology LetX a beatopologicalspaceforeacha? A . 1.In ? X a , thesetsoftheform ? U a , where U a isopeninX a foreacha? A , formabase foratopology. 2.Whatdonhoods f? R R looklikeintheboxtopology?...

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BoxTopology

LetX_abeatopologicalspaceforeacha?A
.

1.In ? X_a
,thesetsoftheform ? U_a
,where U_aisopeninX_a
foreacha?A
,formabase

foratopology.

2.Whatdonhoods f?
R^R
looklikeintheboxtopology? [see8.4(1)].Comparewith

4F3.

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Answered Same DayDec 21, 2021

Answer To: Box Topology LetX a beatopologicalspaceforeacha? A . 1.In ? X a , thesetsoftheform ? U a , where U a...

David answered on Dec 21 2021

121 Votes

1. Let B = {
∏
Uα|Uα open inXα}. We have to prove that this collection
makes the basis for the box topology. For this we have to verify the
following:
(a) As in the collection, we have
∏
Xα ∈ B, Hence B covers
∏
Xα.
(b) Let
{xα} ∈ (
∏
Uα) ∩ (
∏
Vα),
Then we have {xα} ∈
∏
Uα and {xα}...

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Box Topology LetX a beatopologicalspaceforeacha? A . 1.In ? X a , thesetsoftheform ? U a , where U a isopeninX a foreacha? A , formabase foratopology. 2.Whatdonhoods f? R R looklikeintheboxtopology?...

Answer To: Box Topology LetX a beatopologicalspaceforeacha? A . 1.In ? X a , thesetsoftheform ? U a , where U a...

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