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