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Real analysis

A set E is totally disconnected if, given any two points x,y belong to E there exist separated sets A and B with x belong to A and y belong to B and E=A U B.

1-show that Q is totally disconnected.
2-is the set of irrational numbers totally disconnected?

Solution Preview

1. For any x,y in Q and x<y, we can find some irrational number z such that x<z<y. Now let A={x in Q: x<z} and B={x in Q: x>z}, then Q=A U B and A ...

Solution Summary

This is a proof regarding irrational numbers and disconnectedness.