**Problem 4: Determining Open and Closed Sets**For each set, determine whether the following sets are open, closed, or neither. If a set is not open, identify a point in the set for which there is no ε-neighborhood contained within the set. If a set is not closed, find a limit point that is not contained in the set.**Sets to Evaluate:**(a) $ \mathbb{Q} $ (The set of rational numbers)(b) $ \mathbb{N} $ (The set of natural numbers)(c) $ \{ x \in \mathbb{R} : x \neq 0 \} $ (The set of all non-zero real numbers)

OPen set:-S open in ℝ if, for all x∈S, there exist δ>0 such that x-δ,x+δ⊂SSet of rational number…

4. For each set determine whether the following sets are open, closed, or neither. If a set is not open, find a point in the set for which there is no E-neighborhood contained in the set. If a set is not closed, find a limit point that is not contained in the set. (a) Q (b) N (c) {r €R:r+0}

4. For each set determine whether the following sets are open, closed, or neither. If a set is not open, find a point in the set for which there is no E-neighborhood contained in the set. If a set is not closed, find a limit point that is not contained in the set. (a) Q (b) N (c) {r €R:r+0}

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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Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

**Problem 4: Determining Open and Closed Sets**

For each set, determine whether the following sets are open, closed, or neither. If a set is not open, identify a point in the set for which there is no ε-neighborhood contained within the set. If a set is not closed, find a limit point that is not contained in the set.

**Sets to Evaluate:**

(a) $ \mathbb{Q} $ (The set of rational numbers)

(b) $ \mathbb{N} $ (The set of natural numbers)

(c) $ \{ x \in \mathbb{R} : x \neq 0 \} $ (The set of all non-zero real numbers)

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