Question #4 Here is the transcription of the image for use on an educational website:---### Real Analysis and Metric Spaces Problem Set#### 1. Metric DeterminationLet $ 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. Interior Points and Open SetsLet $ E \subseteq X $ where $ X $ is a metric space. Let $ E^\circ $ denote all the interior points of $ E $. $ 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 $ E $ is open if and only if $ E = E^\circ $. - (b) Show: 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 $. Show $ K_1 \cup K_2 $ is compact.#### 4. Closedness of Intersection of Compact SetsLet $\{K_\alpha\}_{\alpha \in A}$ be a collection of compact subsets of a metric space $ X $. Show $\bigcap_{\alpha \in A} K_\alpha$ is closed in $ X $.#### 5. Non-compactness of Open and Compact Sets - (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

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Answered: 4. Let {Ka}aeA be a collection of…

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4. Let {Ka}aeA be a collection of compact subsets of a metric space, Show NaEA K, is closed

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

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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!

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Question #4

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---

### Real Analysis and Metric Spaces Problem Set

#### 1. Metric Determination
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. Interior Points and Open Sets
Let $ E \subseteq X $ where $ X $ is a metric space. Let $ E^\circ $ denote all the interior points of $ E $. $ 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 $ E $ is open if and only if $ E = E^\circ $.
- (b) Show: 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 $. Show $ K_1 \cup K_2 $ is compact.

#### 4. Closedness of Intersection of Compact Sets
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. Non-compactness of Open and Compact Sets
- (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

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