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Intervals with rational endpoints as a basis for topology

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Prove that the family of open intervals with rational end points is a basis for the topology of the real line.
A rigorous proof with detailed explanations is given.

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Solution Summary

We prove that the set of intervals with rational endpoints is the basis for topology on R.

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A collection O of sets is a basis for topology of X if every open set V in X (including X itself) can be represented as a union of sets in O.
The easiest way to show that a collection of subsets if a basis of topology, is to show that relatively to another base.

The metric topology on R is generated by open intervals (x-e,x+e) centered at x of length 2e. If we show ...

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