Technique of decomposing a real interval into intervals

[wayneckm]

Hello all,

I always come across the technique of decomposing a real interval into intervals with rational end point, however, I am a bit confused with the half-open/half-closed cases. For example,

$$[0,t) = \cup_{q < t, q \in \mathbb{Q}} [0,q)$$. And for the case of $$[0,t]$$, we can only construct from using "outer sense", meaning that using all rational $$q > t$$?

Also, what is the set of $$\cup_{q < t, q \in \mathbb{Q}} [0,q]$$?

Thanks.

 

[CompuChip]

Actually I don't think it matters if you take [0, q) or [0, q] in the union, because there will always be a rational number arbitrarily close but smaller than t. In other words, for any $q < t, q \in \mathbb{Q}$ you can always find $q' \in \mathbb{Q}$ such that q < q' < t.

For the case of [0, t] I think you should be taking an intersection, like
$$[0, t] = \cap_{q > t, q \in \mathbb{Q}} [0, q)$$
(or, again, [0, q] will do).
You can check that t will be contained in the intersection (it is in every interval of the form [0, q) with q > t) but no number t' > t is (because you can always find a rational q such that t < q < t').