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Finite complement topology

by Nusc
Tags: complement, finite, topology
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Nusc
#1
Oct26-07, 11:48 PM
P: 776
I need to show if the finite complement topology,T_3, and the topology having all sets (-inf,a) = {x|x<a} as basis ,T_5, are comparable.

I've shown that T_3 is not strictly finer than T_5.

But I'm not sure about other case.

I need help.
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morphism
#2
Oct27-07, 12:09 AM
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I presume you're defining these topologies on R and that you managed to find a set in T_5 that's not in T_3. The other direction is just as easy: is R\{0} in T_5?
Nusc
#3
Oct27-07, 06:17 PM
P: 776
R\{0} is not in T_5. (-inf,0]U[0,inf)

If T_4 is the upper limit topology, having the sets (a,b] as a basis and
T_2 the topology of R_K (a,b)-K K = 1/n n in Z.

I've shown that T_2 is not strictly finer than T_4.
How do I show that T_4 is strictly finer than T_2?

Nusc
#4
Oct27-07, 06:21 PM
P: 776
Finite complement topology

And most importantly, why is R\{1,2,3} considered a basis element for T_3?

It's not finite, nor is it all of R. I'm confused with the definition of a finite complement topology.
Nusc
#5
Oct27-07, 08:13 PM
P: 776
Is {1} = (0,2) ?
and R\{0} = (-inf,0]U[0,inf) ?
Moonbear
#6
Oct27-07, 09:13 PM
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Note: I have merged your two threads on this since you had already received responses in the Calc and Analysis forum.
Nusc
#7
Oct28-07, 11:29 PM
P: 776
Hello?
HallsofIvy
#8
Oct29-07, 06:37 AM
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Thanks
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Look at morphism's post!


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