Question #5 Here is a transcription of the content as it would appear on an educational website, with a detailed explanation of any visual elements:---### Metric Spaces and Compactness Problems1. **Let $ x, y \in \mathbb{R} $. Determine if the following are metrics on $ \mathbb{R} $ or not:** - (a) $ d_1(x, y) = \sqrt{|x - y|} $ - (b) $ d_2(x, y) = (x - y)^4 $2. **Let $ E \subseteq X $ where $ X $ is a metric space.** Let $ E^o $ denote all the interior points of $ E $. $ E^o $ is an open set—you don’t have to prove this, but you can use this fact to prove the following: - (a) Show $ E $ is open if and only if $ E = E^o $. - (b) Show: If $ G \subseteq E $ and $ G $ is open, then $ G \subseteq E^o $.3. **Let $ K_1 $ and $ K_2 $ be compact subsets of a metric space $ X $.** Show $ K_1 \cup K_2 $ is compact.4. **Let $ \{K_\alpha\}_{\alpha \in A} $ be a collection of compact subsets of a metric space $ X $.** Show $ \bigcap_{\alpha \in A} K_\alpha $ is closed in $ X $.5. **(a) Consider the collection of open sets, $ \{(-n, n)\}_{n=1}^\infty $ in $ \mathbb{R} $ with the usual metric $ d(x, y) = |x - y| $.** Use this collection of open sets to show $ \mathbb{R} $ is not compact (make sure to prove $ \mathbb{R} = \bigcup_{n \in \mathbb{N}} (-n, n) $ as part of your work). - (b) Give an example of a collection of compact sets in $ \mathbb{R} $, say \( \{K_n\}_{n

Answered: Image /qna-images/answer/4582ed71-69d4-4a29-9d2c-5b420d15b881.jpg

^2 be pact sub els of a tnc space A. Show NaEA Ka is closed 4. Let {Ka}aea be a collection of compact subsets of a metric space, in X 5. (a) Consider the collection of open sets, {(-n,n)}"=1 in R with the usual metric d(x, y) = |x – y| . Use this collection of open sets to show R is not compact (make sure to prove R = Unej(-n, n) as part of your work). (b) Give an example of a collection of compact sets in R , say {Kn}n=1 such that U=1 Kn is not compact (Hint: consider part (a) ) 6. Let E C X where X is a metric space, Show the limit points of E are the same as the limit

^2 be pact sub els of a tnc space A. Show NaEA Ka is closed 4. Let {Ka}aea be a collection of compact subsets of a metric space, in X 5. (a) Consider the collection of open sets, {(-n,n)}"=1 in R with the usual metric d(x, y) = |x – y| . Use this collection of open sets to show R is not compact (make sure to prove R = Unej(-n, n) as part of your work). (b) Give an example of a collection of compact sets in R , say {Kn}n=1 such that U=1 Kn is not compact (Hint: consider part (a) ) 6. Let E C X where X is a metric space, Show the limit points of E are the same as the limit

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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Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!

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Question #5

Here is a transcription of the content as it would appear on an educational website, with a detailed explanation of any visual elements:

---

### Metric Spaces and Compactness Problems

1. **Let $ x, y \in \mathbb{R} $. Determine if the following are metrics on $ \mathbb{R} $ or not:**
- (a) $ d_1(x, y) = \sqrt{|x - y|} $
- (b) $ d_2(x, y) = (x - y)^4 $

2. **Let $ E \subseteq X $ where $ X $ is a metric space.** Let $ E^o $ denote all the interior points of $ E $. $ E^o $ is an open set—you don’t have to prove this, but you can use this fact to prove the following:
- (a) Show $ E $ is open if and only if $ E = E^o $.
- (b) Show: If $ G \subseteq E $ and $ G $ is open, then $ G \subseteq E^o $.

3. **Let $ K_1 $ and $ K_2 $ be compact subsets of a metric space $ X $.** Show $ K_1 \cup K_2 $ is compact.

4. **Let $ \{K_\alpha\}_{\alpha \in A} $ be a collection of compact subsets of a metric space $ X $.** Show $ \bigcap_{\alpha \in A} K_\alpha $ is closed in $ X $.

5. **(a) Consider the collection of open sets, $ \{(-n, n)\}_{n=1}^\infty $ in $ \mathbb{R} $ with the usual metric $ d(x, y) = |x - y| $.** Use this collection of open sets to show $ \mathbb{R} $ is not compact (make sure to prove $ \mathbb{R} = \bigcup_{n \in \mathbb{N}} (-n, n) $ as part of your work).
- (b) Give an example of a collection of compact sets in $ \mathbb{R} $, say \( \{K_n\}_{n

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