I Need Guide To Comprehend This Counter Example Of A Not Compat Set

Introduction

In the realm of topology, compactness is a fundamental property that has far-reaching implications in various areas of mathematics. A compact set is one that has the property that every open cover has a finite subcover. However, not all sets possess this property, and understanding the counterexamples is crucial for grasping the concept of compactness. In this article, we will delve into a specific counterexample of a non-compact set, namely the set $S_{ℓ_{∞}}=\{x=(x_{n})∈ℓ_{∞}:∥x∥_{∞}=1\}$ in the space $ℓ_{∞}$.

The Space $ℓ_{∞}$

Before we proceed, let us briefly recall the definition of the space $ℓ_{∞}$. The space $ℓ_{∞}$ consists of all bounded sequences of real numbers, equipped with the norm $∥x∥_{∞}=\sup_{n∈ℕ}|x_{n}|$. In other words, $ℓ_{∞}$ is the set of all sequences $x=(x_{n})$ such that $|x_{n}|≤M$ for some real number $M$ and all $n∈ℕ$. The norm $∥⋅∥_{∞}$ is defined as the supremum of the absolute values of the terms of the sequence.

The Set $S_{ℓ_{∞}}$

The set $S_{ℓ_{∞}}$ is defined as the subset of $ℓ_{∞}$ consisting of all sequences $x=(x_{n})$ such that $∥x∥_{∞}=1$. In other words, $S_{ℓ_{∞}}$ is the set of all sequences with norm equal to 1. This set is often referred to as the unit sphere in $ℓ_{∞}$.

The Open Cover

To show that $S_{ℓ_{∞}}$ is not compact, we consider the open cover $\{B(x,1/2):x∈S_{ℓ_{∞}}\}$. This open cover consists of all open balls of radius 1/2 centered at each point of $S_{ℓ_{∞}}$. In other words, for each $x∈S_{ℓ_{∞}}$, we have the open ball $B(x,1/2)=\{y∈ℓ_{∞}:∥y-x∥_{∞}<1/2\}$.

The Counterexample

To show that this open cover does not have a finite subcover, we need to find a sequence of points in $S_{ℓ_{∞}}$ such that no finite subcollection of the open balls in the cover contains all these points. Let us consider the sequence of points $x^{(n)}=(x_{n}^{(n)})$ defined by $x_{n}^{(n)}=1$ if $n=k$ and $x_{n}^{(n)}=0$ otherwise. This sequence is clearly in $S_{ℓ_{∞}}$ since $∥x^{(n)}∥_{∞}=1$ for all $n∈ℕ$.

The Finite Sub

Suppose, for the sake of contradiction, that the open cover $\{B(x,1/2):x∈S_{ℓ_{∞}}\}$ has a finite subcover. Then there exists a finite set of points $x^{(1)},x^{(2)},...,x^{(m)}$ in $S_{ℓ_{∞}}$ such that the open balls $B(x^{(1)},1/2),B(x^{(2)},1/2),...,B(x^{(m)},1/2)$ cover all the points in the sequence $x^{(n)}$. However, this is impossible since the points $x^{(n)}$ are all at a distance of at least 1/2 from each other. Indeed, suppose that $x^{(n)}$ and $x^{(k)}$ are two points in the sequence such that $n≠k$. Then $x_{n}^{(n)}=1$ and $x_{k}^{(n)}=0$, so $∥x^{(n)}-x^{(k)}∥_{∞}=1$. This implies that the open balls $B(x^{(n)},1/2)$ and $B(x^{(k)},1/2)$ do not intersect, and therefore the points $x^{(n)}$ and $x^{(k)}$ are not both contained in the finite subcover.

Conclusion

We have shown that the set $S_{ℓ_{∞}}$ is not compact by exhibiting an open cover that does not have a finite subcover. This counterexample highlights the importance of understanding the properties of compact sets and the need to be able to recognize and construct counterexamples.

Compactness and Its Applications

Compactness is a fundamental property in topology that has far-reaching implications in various areas of mathematics. It is used to prove the existence of solutions to equations, the convergence of sequences, and the continuity of functions. Understanding compactness is essential for working in many areas of mathematics, including real analysis, functional analysis, and differential equations.

Compactness in Real Analysis

In real analysis, compactness is used to prove the existence of solutions to equations. For example, the Bolzano-Weierstrass theorem states that every bounded sequence of real numbers has a convergent subsequence. This theorem is a consequence of the compactness of the closed unit interval $[0,1]$.

Compactness in Functional Analysis

In functional analysis, compactness is used to prove the existence of solutions to equations. For example, the Arzelà-Ascoli theorem states that every bounded sequence of continuous functions on a compact set has a subsequence that converges uniformly. This theorem is a consequence of the compactness of the space of continuous functions on a compact set.

Compactness in Differential Equations

In differential equations, compactness is used to prove the existence of solutions to equations. For example, the Cauchy-Lipschitz theorem states that every initial value problem for a differential equation has a unique solution. This theorem is a consequence of the compactness of the space of solutions to the differential equation.

Conclusion

Introduction

In our previous article, we explored the concept of compactness and its applications in various areas of mathematics. We also examined a counterexample of a non-compact set, namely the set $S_{ℓ_{∞}}$ in the space $ℓ_{∞}$. In this article, we will address some frequently asked questions about compactness and its applications.

Q: What is compactness?

A: Compactness is a fundamental property in topology that has far-reaching implications in various areas of mathematics. A compact set is one that has the property that every open cover has a finite subcover.

Q: Why is compactness important?

A: Compactness is important because it allows us to prove the existence of solutions to equations, the convergence of sequences, and the continuity of functions. It is used in many areas of mathematics, including real analysis, functional analysis, and differential equations.

Q: What is an open cover?

A: An open cover is a collection of open sets that cover a given set. In other words, it is a collection of open sets such that every point in the given set is contained in at least one of the open sets.

Q: What is a finite subcover?

A: A finite subcover is a finite collection of open sets that cover a given set. In other words, it is a finite collection of open sets such that every point in the given set is contained in at least one of the open sets.

Q: How do you prove that a set is compact?

A: To prove that a set is compact, you need to show that every open cover has a finite subcover. This can be done by using various techniques, such as the Bolzano-Weierstrass theorem, the Arzelà-Ascoli theorem, or the Cauchy-Lipschitz theorem.

Q: What is the Bolzano-Weierstrass theorem?

A: The Bolzano-Weierstrass theorem states that every bounded sequence of real numbers has a convergent subsequence. This theorem is a consequence of the compactness of the closed unit interval $[0,1]$.

Q: What is the Arzelà-Ascoli theorem?

A: The Arzelà-Ascoli theorem states that every bounded sequence of continuous functions on a compact set has a subsequence that converges uniformly. This theorem is a consequence of the compactness of the space of continuous functions on a compact set.

Q: What is the Cauchy-Lipschitz theorem?

A: The Cauchy-Lipschitz theorem states that every initial value problem for a differential equation has a unique solution. This theorem is a consequence of the compactness of the space of solutions to the differential equation.

Q: Can you provide more examples of compact sets?

A: Yes, here are a few more examples of compact sets:

• The closed unit interval $[0,1]$ in the real numbers
• The closed unit ball in the Euclidean space $\mathbb{R}^n$
• The closed unit sphere in the Euclidean space \mathbb{R^n
• The set of all continuous functions on a compact set
• The set of all bounded sequences of real numbers

Q: Can you provide more examples of non-compact sets?

A: Yes, here are a few more examples of non-compact sets:

• The set of all real numbers
• The set of all rational numbers
• The set of all irrational numbers
• The set of all bounded sequences of real numbers that do not converge
• The set of all continuous functions on a non-compact set

Conclusion

In conclusion, compactness is a fundamental property in topology that has far-reaching implications in various areas of mathematics. Understanding compactness is essential for working in many areas of mathematics, including real analysis, functional analysis, and differential equations. We hope that this Q&A article has provided a helpful overview of compactness and its applications.