Explore BrainMass

Explore BrainMass

    Intervals with rational endpoints as a basis for topology

    Not what you're looking for? Search our solutions OR ask your own Custom question.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    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.

    © BrainMass Inc. brainmass.com March 4, 2021, 10:39 pm ad1c9bdddf

    Solution Preview

    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 ...

    Solution Summary

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