How to prove Sorgenfrey topology stronger than standard?

Question

I understand that my title's statement is true and I'm not sure why. Any [a,b) set is open on Sorgenfrey's so I guess I could use a union of all [a + 1/n, b) sets in order to get the (a,b) set which is open on the standard topology.

Is that correct?

But, if this is true, then why not union all [a, b+1/n) sets in order to get the set [a,b] and thus prove [a,b] is also open on Sorgenfrey topology. I'm confused...
thx

Answer 1

The set $[a,b]$ is equal to the intersection $$[a,b] =\bigcap_n \left[ a, b+\frac{1}{n}\right)$$ not the union $$\bigcup_n \left[ a, b+\frac{1}{n}\right) .$$