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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,

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

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

Thanks.