S.show that in the metric space (R, I.1),Q = 4, (Q')'=%3DCinteripr of rational numbers and interior of irrationalnumbers'are empty set)

Answered: s Show that in the metric space (R,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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S.
show that in the metric space (R, I.1),
Q = 4, (Q')'=
%3D
Cinteripr of rational numbers and interior of irrational
numbers'
are empty set)

Expert Solution

Step 1

A rational number x is called interior point if there exists a neighbourhood of x which contains only points from rational numbers.

But rational are dense in real with respect to this metric,so between any two rational number there is always a irrational number ,so such neighbourhood does not exists.

So for any rational number x, x is not an interior point .

Hence interior is empty set.

Similarly we can prove that for irrational numbers.

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