Basis for the Lower Limit Topology?

We show that the intervals with rational endpoints [a,b) form a basis for a topology on the real line. Then we show the topology generated by these intervals is not the lower limit topology. In other words, using intervals [a,b) with rational endpoints as the building blocks yields a different topology than using intervals [a,b) with real endpoints.

[mohammadamanalimyzada]

Hello sir.
I have got a small doubt regarding if the interval [a, b) can cover all the real line. The left side is close so my doubt is if we can include the minus infinity in the interval. in other words, can we write [ -infinity, b)?

greatly appreciated in advance