Problem 10 - Basic Analysis: Introduction to Real Analysis

Question

a) Working in \(\mathbb{R}\), compute \(\operatorname{diam}([a, b])\). b) Working in \(\mathbb{R}^{n},\) for any \(r>0,\) let \(B_{r}:=\left\\{x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}

Step-by-Step Solution

Answer

a) \(\operatorname{diam}([a,b]) = |b-a|\); b) \(\operatorname{diam}(B_r) = 2r\); c) \(\operatorname{diam}(\{p\}) = 0\), \(\operatorname{diam}(X) = 1\), and \((X, d)\) is bounded.

Step 1: Understanding the Diameter in Real Numbers for Interval [a, b]

The diameter of a set in real numbers, specifically an interval \([a, b]\), is the distance between the furthest points in the set. In this case, that would be the endpoints \(a\) and \(b\).

Step 2: Calculating Diameter for [a, b]

The formula for the distance between two points \(a\) and \(b\) in real numbers is \(|b - a|\) (absolute value of \(b - a\). Therefore: \[\operatorname{diam}([a, b]) = |b - a|\]

Step 3: Understanding the Diameter in Real Numbers for Open Ball in \(\mathbb{R}^n\)

The set \(B_{r}\) is defined as all points \((x_1, x_2, \ldots, x_n)\) with the inequality \(x_1^2 + x_2^2 + \cdots + x_n^2

Step 4: Determining Diameter of \(B_r\)

The diameter of \(B_{r}\) equates to twice the radius since every point in\(B_{r}\) can lie within a distance of the radius \(r\) from the center. Hence\[ \operatorname{diam}(B_r) = 2r \]

Step 5: Explaining the Discrete Metric for \(\{p\}\) and \(X\)

In a metric space with a discrete metric, the diameter of a single point set \(\{p\}\) is zero because the only elements are equal (\(p\)). As for \(X\), because \(d(x,y) = 1\) for distinct points \(x, y\), the diameter here will highlight the greatest possible distance between two distinct points.

Step 6: Calculating Diameters Using the Discrete Metric

Thus, \(\operatorname{diam}(\{p\}) = 0\). Also,\(\operatorname{diam}(X) = 1\), illustrating that the greatest distance between any two distinct points is 1.

Step 7: Concluding Boundedness in the Metric Space \((X, d)\)

A metric space is bounded if there exists a real number \(M\) such that the distance between any two points is less than \(M\). Here, the maximum distance is 1, thus \((X, d)\) is bounded.

Key Concepts

Diameter

The concept of diameter in a metric space helps us understand the extent or size of the set by measuring the maximum distance between any two points within the set. In simpler terms, it's like asking how wide the set is from the furthest apart points it contains.
When you have a set, such as an interval \[a, b\] in real numbers \(\mathbb{R}\), the diameter is straightforward to calculate. You simply measure the absolute value of the difference between its endpoints \(a\) and \(b\). Thus, the formula for the diameter of an interval \([a, b]\) is given by \(\operatorname{diam}([a, b]) = |b - a|\).
This provides an easy and intuitive way to apprehend the concept of distance in a one-dimensional scenario.

Discrete Metric

A discrete metric is an intriguing concept where the distance function is defined in a very particular way, simplifying certain analyses. In this metric space, the distance \(d(x,y)\) between any two distinct points \(x\) and \(y\) is always 1. Meanwhile, the distance from a point to itself is always 0.
This leads to some interesting results when examining the diameter of sets. For a single-point set \({p}\), its diameter will naturally be 0, as there are no different points to measure a distance between.
However, when considering the entire set \(X\), which contains two distinct points at least, the diameter is 1. This is because there is always a pair of distinct points, the maximum distance being exactly parameterized by our definition of the discrete metric. This illustrates a simpler snapshot of distance measure but within the confined rules of the discrete metric definition.

Open Ball

Imagine an open ball in the context of spaces \(\mathbb{R}^n\). It's a set defined by taking a center point and assembling all nearby points within a certain radius, not including the edge itself.
If we have an open ball \(B_r\) in \(\mathbb{R}^n\), defined as those points \((x_1, x_2, \ldots, x_n)\) such that \(x_1^2 + x_2^2 + \cdots + x_n^2 < r^2\), the diameter is intriguing. Here, the diameter corresponds to twice the radius, \(2r\). This is because, by moving out from the center, any point is only at most a single radius away, giving the diameter as the total span across the open ball.
The open ball thus presents a fundamental and valuable visualization for understanding distance in more dimensions.

Boundedness

Boundedness is a term that reflects whether a set can be contained within a ball of finite radius in a metric space. If this can be accomplished, the set is deemed bounded.
In metric spaces like \((X, d)\), when equipped with a discrete metric, this is showcased effectively. The fact that the greatest distance between any two distinct points in \(X\) is simply 1 demonstrates the property of boundedness quite emphatically. If you consider any possible two points, they can never have a distance greater than this set maximum value.
Thus, with such a metric space, you can always choose a number greater than 1 as a boundary, attesting to the notion that the space is bounded. This neatly boxes in the behavior of distances, ensuring a finite measure that's pivotal in many mathematical analyses.

Previous

Problem 10

Next

Problem 11

Other exercises in this chapter

Problem 10

Suppose \((X, d)\) is complete and suppose we have a countably infinite collection of nonempty compact sets \(E_{1} \supset E_{2} \supset E_{3} \supset \cdots .

View solution

Problem 10

Take \(\mathbb{R}^{*}=\\{-\infty\\} \cup \mathbb{R} \cup\\{\infty\\}\) be the extended reals. Define \(d(x, y):=\left|\frac{x}{1+|x|}-\frac{y}{1+|y|}\right|\) i

View solution

Problem 11

Let \(A\) be a connected set in a metric space. a) Is \(\bar{A}\) connected? Prove or find a counterexample. b) Is A \(^{\circ}\) connected? Prove or find a cou

View solution

Problem 11

Suppose \(\left\\{V_{n}\right\\}_{n=1}^{\infty}\) is a collection of open sets in \((X, d)\) such that \(V_{n+1} \supset V_{n} .\) Let \(\left\\{x_{n}\right\\}\

View solution

Key Concepts

Diameter

The concept of diameter in a metric space helps us understand the extent or size of the set by measuring the maximum distance between any two points within the set. In simpler terms, it's like asking how wide the set is from the furthest apart points it contains.
When you have a set, such as an interval \[a, b\] in real numbers \(\mathbb{R}\), the diameter is straightforward to calculate. You simply measure the absolute value of the difference between its endpoints \(a\) and \(b\). Thus, the formula for the diameter of an interval \([a, b]\) is given by \(\operatorname{diam}([a, b]) = |b - a|\).
This provides an easy and intuitive way to apprehend the concept of distance in a one-dimensional scenario.

Discrete Metric

A discrete metric is an intriguing concept where the distance function is defined in a very particular way, simplifying certain analyses. In this metric space, the distance \(d(x,y)\) between any two distinct points \(x\) and \(y\) is always 1. Meanwhile, the distance from a point to itself is always 0.
This leads to some interesting results when examining the diameter of sets. For a single-point set \({p}\), its diameter will naturally be 0, as there are no different points to measure a distance between.
However, when considering the entire set \(X\), which contains two distinct points at least, the diameter is 1. This is because there is always a pair of distinct points, the maximum distance being exactly parameterized by our definition of the discrete metric. This illustrates a simpler snapshot of distance measure but within the confined rules of the discrete metric definition.

Open Ball

Imagine an open ball in the context of spaces \(\mathbb{R}^n\). It's a set defined by taking a center point and assembling all nearby points within a certain radius, not including the edge itself.
If we have an open ball \(B_r\) in \(\mathbb{R}^n\), defined as those points \((x_1, x_2, \ldots, x_n)\) such that \(x_1^2 + x_2^2 + \cdots + x_n^2 < r^2\), the diameter is intriguing. Here, the diameter corresponds to twice the radius, \(2r\). This is because, by moving out from the center, any point is only at most a single radius away, giving the diameter as the total span across the open ball.
The open ball thus presents a fundamental and valuable visualization for understanding distance in more dimensions.

Boundedness

Boundedness is a term that reflects whether a set can be contained within a ball of finite radius in a metric space. If this can be accomplished, the set is deemed bounded.
In metric spaces like \((X, d)\), when equipped with a discrete metric, this is showcased effectively. The fact that the greatest distance between any two distinct points in \(X\) is simply 1 demonstrates the property of boundedness quite emphatically. If you consider any possible two points, they can never have a distance greater than this set maximum value.
Thus, with such a metric space, you can always choose a number greater than 1 as a boundary, attesting to the notion that the space is bounded. This neatly boxes in the behavior of distances, ensuring a finite measure that's pivotal in many mathematical analyses.