1. ## Questions on topology

Hi, I've got some questions about topologies.

1) Would B ={ [a,b] | a $\displaystyle \leq$ b}, where a and b are real, be a basis for the discrete topology on the reals (under the open set definition of topology)?

2) Is the box topology always strictly finer than the product topology whenever the index set is infinity?

For 1) I think it is ... but I might have missed something

For 2) I don't think that's the case, my counter example would be if we have a finite index set (so the box and product topologies are the same), then we could extend it to an infinite set where the new open sets are simply equal to the original set (i.e. U = X, for all the new open sets)

Just wanted to check if my understanding of these things are correct .... or if I should start freaking out now hehehe

2. ## Re: Questions on topology

1) You're right. What are you worried you might have missed? Or, why do you think it's discrete?

2) I'm not sure I understand your argument. What do you mean by "extend it to an infinite set where the new open sets are simply equal to the original set"? Extend what?

A hint on (2): A basis element for the box topology is a product of a nonempty open set from each factor space. To also be a basis element for the product topology, all but finitely many of those open sets need to be the entire space in that coordinate. How can you make those coincide?