Comparing Finite Complement and (-inf,a) Topologies: Are T_3 and T_5 Comparable?

[Nusc]

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.

 

[morphism]

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]

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]

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]

Topology

Is {1} = (0,2) ?
and R\{0} = (-inf,0]U[0,inf) ?

 

[Moonbear]

Note: I have merged your two threads on this since you had already received responses in the Calc and Analysis forum.

 

[Nusc]

Hello?

 

[HallsofIvy]

Look at morphism's post!