product of compact sets compact in box topology?


by spicychicken
Tags: compact, product, sets, topology
spicychicken
spicychicken is offline
#1
Jul20-11, 11:33 AM
P: 3
So Tychonoff theorem states products of compact sets are compact in the product topology.

is this true for the box topology? counterexample?
Phys.Org News Partner Science news on Phys.org
Cougars' diverse diet helped them survive the Pleistocene mass extinction
Cyber risks can cause disruption on scale of 2008 crisis, study says
Mantis shrimp stronger than airplanes
micromass
micromass is offline
#2
Jul21-11, 08:08 AM
Mentor
micromass's Avatar
P: 16,690
A counterexample is [itex]\prod_{n\in \mathbb{N}}{[0,1]}[/itex]. Can you show why?
spicychicken
spicychicken is offline
#3
Jul21-11, 01:17 PM
P: 3
if S_n is the set with empty sets in each index except n where for index n you have [0,1], then {S_n} is an open cover with no finite subcover...i think

micromass
micromass is offline
#4
Jul21-11, 02:24 PM
Mentor
micromass's Avatar
P: 16,690

product of compact sets compact in box topology?


Quote Quote by spicychicken View Post
if S_n is the set with empty sets in each index except n where for index n you have [0,1], then {S_n} is an open cover with no finite subcover...i think
Such a sets will always be empty. Try to consider a cover by all sets of the form

[tex]\prod_{n\in \mathbb{N}}{A_i}[/tex]

Where Ai=[0,0.6[ or Ai=]0.5,1]


Register to reply

Related Discussions
[Topology]Determining compact sets of R Calculus & Beyond Homework 4
Topology compact sets Calculus & Beyond Homework 4
How can you prove that a Cartesian product of compact sets is compact? Calculus 2
PROVING INTERSECTION OF Any number of COMPACT SETS is COMPACT? Calculus 4
[SOLVED] Topology: Nested, Compact, Connected Sets Calculus & Beyond Homework 5