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Definition. A topological space X is separable if and only if X has a countable dense subset. Exercise 5.2. Show that Rstd is separable. With which of the topologies on R that you have studied is R not separable?

Definition. A topological space X is separable if and only if X has a countable dense subset. Exercise 5.2. Show that Rstd is separable. With which of the topologies on R that you have studied is R not separable?

Advanced Engineering Mathematics
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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Could you explain how to show 5.2 in easiest possible way(in very detail)?

**Definition.** A topological space \( X \) is **separable** if and only if \( X \) has a countable dense subset. **Exercise 5.2.** Show that \( \mathbb{R}_{\text{std}} \) is separable. With which of the topologies on \( \mathbb{R} \) that you have studied is \( \mathbb{R} \) not separable?
Transcribed Image Text:**Definition.** A topological space \( X \) is **separable** if and only if \( X \) has a countable dense subset. **Exercise 5.2.** Show that \( \mathbb{R}_{\text{std}} \) is separable. With which of the topologies on \( \mathbb{R} \) that you have studied is \( \mathbb{R} \) not separable?
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