Answered: Find the interior, exterior, boundary,…

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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Question

Hello there, can you help me solve a problem?

Find the interior, exterior, boundary, closure, accumulation pts, and isolated pts of the following subset of R. Justify your answer.

Thanks!

Expert Solution

Step 1: For Interior set

The point x is called an interior point of the set E if for some ε > 0 the entire ε-neighborhood (x − ε, x + ε) is contained in E.

If there p is a point in set $open curly brackets r element of Q colon 0 less than r less than straight pi close curly brackets$ such that there is some $element of negative$ neighbourhood of p wholly lie in set $open curly brackets r element of Q colon 0 less than r less than straight pi close curly brackets$

Then, as $a n y space element of negative n e i g h b o u r h o o d$ of p must contain irrational numbers, that implies by above, irrational numbers must be in the set $open curly brackets r element of Q colon 0 less than r less than straight pi close curly brackets$ , a contradiction.

Hence, there is no interior point of the set $open curly brackets r element of Q colon 0 less than r less than straight pi close curly brackets$

So, required interior set is empty.