Question #7 ---## Metrics and Compactness in Metric Spaces### 1. Determine if the following functions are metrics on $\mathbb{R}$:Given $x, y \in \mathbb{R}$: **(a)** $d_1(x, y) = \sqrt{|x - y|}$ **(b)** $d_2(x, y) = (x - y)^4$### 2. Interior Points and OpennessLet $E \subseteq X$ where $X$ is a metric space. Let $E^\circ$ denote all the interior points of $E$. **Note:** $E^\circ$ is an open set—you don’t have to prove this, but you can use this fact to prove the following.#### (a) Show that $E$ is open if and only if $E = E^\circ$.#### (b) Prove that if $ G \subseteq E $ and $ G $ is open, then $ G \subseteq E^\circ $.### 3. Compact SubsetsLet $ K_1 $ and $ K_2 $ be compact subsets of a metric space $X$. Prove that $ K_1 \cup K_2 $ is compact.### 4. Collection of Compact SubsetsLet $ \{ K_\alpha \}_{\alpha \in A} $ be a collection of compact subsets of a metric space $X$. Show that $ \cap_{\alpha \in A} K_\alpha $ is closed in $X$.### 5. Open Sets and Compactness in $\mathbb{R}$#### (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 $\mathbb{R} = \cup_{n \in \mathbb{N}} (-n, n)$ as part of your work.#### (b) Give an example of a collection of compact sets in $\mathbb{R}$, say \( \{K

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Answered: 7. (a) Let A C X,B C X where X is a…

7. (a) Let A C X,B C X where X is a metric space. Show AnB CĀ OĒ. (b) Consider A = [3,7),B = (7,9] as subset of IR with the usual metric, dr (x, y) = | x – y| . Show Ā n B is not a subset of AnB

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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Concept explainers

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!

Question

Question #7

---

## Metrics and Compactness in Metric Spaces

### 1. Determine if the following functions are metrics on $\mathbb{R}$:

Given $x, y \in \mathbb{R}$:
**(a)** $d_1(x, y) = \sqrt{|x - y|}$
**(b)** $d_2(x, y) = (x - y)^4$

### 2. Interior Points and Openness

Let $E \subseteq X$ where $X$ is a metric space. Let $E^\circ$ denote all the interior points of $E$.
**Note:** $E^\circ$ is an open set—you don’t have to prove this, but you can use this fact to prove the following.

#### (a) Show that $E$ is open if and only if $E = E^\circ$.

#### (b) Prove that if $ G \subseteq E $ and $ G $ is open, then $ G \subseteq E^\circ $.

### 3. Compact Subsets

Let $ K_1 $ and $ K_2 $ be compact subsets of a metric space $X$. Prove that $ K_1 \cup K_2 $ is compact.

### 4. Collection of Compact Subsets

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

### 5. Open Sets and Compactness in $\mathbb{R}$

#### (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 $\mathbb{R} = \cup_{n \in \mathbb{N}} (-n, n)$ as part of your work.

#### (b) Give an example of a collection of compact sets in $\mathbb{R}$, say \( \{K

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