/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 10 On \(\mathbb{R}\) we consider th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 10

On \(\mathbb{R}\) we consider the equivalence relation \(\sim\) given by \(x \sim y\) if \(x-y \in \mathbb{Z}\). Describe the quotient space and the quotient topology.

Short Answer

Expert verified
The quotient space is the circle \(S^1\), and the topology is the quotient topology from \(\mathbb{R}\).

Step by step solution

01

Understanding the Equivalence Relation

The equivalence relation \(x \sim y\) on \(\mathbb{R}\) is defined such that \(x - y \in \mathbb{Z}\). This means that two real numbers \(x\) and \(y\) are equivalent if their difference is an integer. In other words, \(x\) and \(y\) can be considered equal under this relation if they differ by a whole number.
02

Identify Equivalence Classes

An equivalence class of a real number \(x\) under the relation \(\sim\) is the set of all real numbers that are equivalent to \(x\). This can be expressed as: \([x] = \{y \in \mathbb{R} \,|\, y - x \in \mathbb{Z} \}\). This means \([x]\) consists of all numbers that differ from \(x\) by an integer, effectively forming the set \(x + \mathbb{Z}\).
03

Describe the Quotient Space

The quotient space \(\mathbb{R} / \sim\) is the set of these equivalence classes. Each class \([x]\) can be identified with a unique point in the interval \([0,1)\). Thus, the quotient space is the circle \(\mathbb{R}/\mathbb{Z}\), which can be represented as \(S^1 = \{ e^{2\pi ix} \mid x \in [0,1)\} \).
04

Understanding the Quotient Topology

The quotient topology on \(\mathbb{R} / \sim\) is derived from the standard topology on \(\mathbb{R}\) by defining a subset \(U\) of \(\mathbb{R} / \mathbb{Z}\) to be open if and only if its preimage under the natural projection \(\mathbb{R} \rightarrow \mathbb{R} / \mathbb{Z}\) is open in \(\mathbb{R}\). This creates a topology on the circle \(S^1\), making it a commonly considered space in mathematics.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equivalence Relation
An equivalence relation is a way to group elements of a set based on a particular criterion that defines when two elements are considered equivalent. In mathematics, an equivalence relation must satisfy three properties: reflexivity, symmetry, and transitivity.
  • Reflexivity: Any element is equivalent to itself. For every element \(x\), \(x \sim x\).

  • Symmetry: If one element is equivalent to another, then that second element is equivalent to the first. For every pair of elements \(x\) and \(y\), if \(x \sim y\), then \(y \sim x\).

  • Transitivity: If an element is equivalent to a second element, and that second element is equivalent to a third, then the first and third elements are also equivalent. For all elements \(x\), \(y\), and \(z\), if \(x \sim y\) and \(y \sim z\), then \(x \sim z\).

In our specific case, the equivalence relation \(x \sim y\) for real numbers \(x\) and \(y\) is determined by whether their difference is an integer. This sets the ground for grouping all numbers that differ by a whole number together, providing a new way of viewing the set of real numbers.
Equivalence Classes
Once an equivalence relation is defined, it naturally groups a set into distinct subsets called equivalence classes. Each class contains elements that are considered equivalent under the relation.
  • The equivalence class of an element \(x\) under relation \(\sim\) is denoted \([x]\) and is composed of all elements that are equivalent to \(x\).

  • For the equivalence relation \(x \sim y\) (where \(x-y \in \mathbb{Z}\)), the equivalence class \([x]\) consists of all real numbers \(y\) such that \(y - x\) is an integer.

So, for any real number \(x\), its equivalence class is the set \(x + \mathbb{Z}\). This means every number is grouped with numbers like itself within intervals of 1 unit. Equivalence classes under this relation can be visualized on the set of integers which repeats itself throughout the real line.
Topology
Topology is a branch of mathematics concerned with properties of space that are preserved under continuous transformations. It focuses on notions like open and closed sets, continuity, and convergence. Topologically, we think about spaces and how they can be "transformed" without tearing or fitting them in a continuous manner.
  • When a topology is placed on a set, a collection of subsets deemed open forms the topological structure. In considering a quotient space, the topology of the original space \(\mathbb{R}\) influences the topology of its quotient.

  • The set \(\mathbb{R}/\mathbb{Z}\) in itself represents a circle, often denoted as \(S^1\).
The quotient topology on this circle is derived from the standard topology of the real numbers, using a projection that maps each real number to its equivalence class. This provides a very intuitive way to construct a continuous circle from the real line.
Real Numbers
Real numbers encompass all the numbers that can be located on the continuous numerical line, including integers, rational numbers, and irrational numbers. They form a foundational part of mathematics used to represent quantity and continuous space.
  • Integers : Whole numbers that can be positive or negative, and zero.

  • Rational Numbers: Numbers expressible as fractions, which include integers and decimals that repeat or terminate.

  • Irrational Numbers: Numbers that cannot be written as simple fractions, like the square root of 2 or \(\pi\), with non-repeating decimal forms.

Understanding real numbers and how they are tied together with equivalence relations and classes allows for a more profound grasp of topological manipulation and the formation of spaces like the unit circle \(S^1\). The real numbers work together with these concepts to model continuous phenomena in both mathematical theory and real-world applications.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

(The Cantor set.) Let \(C\) denote the set of real numbers \(x\) of the form $$ x=\sum_{n=1}^{\infty} \alpha_{n} 3^{-n}, \quad \text { where } \alpha_{n}=0 \text { or } \alpha_{n}=2 $$ Show that \(C=\bigcap C_{n}\), where \(C_{1}=[0,1]\) and where \(C_{n+1}\) is obtained from \(C_{n}\) by deleting the (open) middle third of each of the intervals that belong to \(C_{n}\). Deduce from this that \(C\) (in the relative topology induced from \(\mathbb{R}\) ) is a compact Hausdorff space. Show that \(C\) as a subset of \(\mathbb{R}\) has empty interior, but no isolated points. Show that every \(x\) in \(C\) has a unique expression \(x=\sum \alpha_{n} 3^{-n}\) and that \(C\) is homeomorphic with the product space \(\\{0,1\\}^{N}\) (and thus uncountable). Show that the map \(f: C \rightarrow[0,1]\) given by \(f(x)=\sum \alpha_{n} 2^{-n-1}\), where \(x=\sum \alpha_{n} 3^{-n}\), is continuous and surjective.

(The Sorgenfrey plane.) Give the set \(\mathbb{R}^{2}\) the topology \(\tau^{2}\), for which a basis consist of products of half-open intervals \(\left[y_{1}, z_{1}\left[\times\left[y_{2}, z_{2}[\right.\right.\right.\), where \(y_{1}, y_{2}, z_{1}\), and \(z_{2}\) range over \(\mathbb{R}\). Show that \(\left(\mathbb{R}^{2}, \tau^{2}\right)\) is a separable space. Show that the subset \(\left\\{(x, y) \in \mathbb{R}^{2} \mid x+y=0\right\\}\) is discrete in the relative topology (and thus nonseparable), but closed in \(\mathbb{R}^{2}\).

(The fundamental group.) Let \((X, \tau)\) be a nonempty arcwise connected (cf. E 1.4.14) topological space, and choose a base point \(x_{0}\) in \(X\). A loop in \(X\) is a continuous function (curve) \(f:[0,1] \rightarrow X\) such that \(f(0)=f(1)=x_{0}\). On the space \(L(X)\) of loops we define a composition \(f g\) (product) by $$ f g(t)=g(2 t), \quad 0 \leq t \leq \frac{1}{2} ; \quad f g(t)=f(2 t-1), \quad \frac{1}{2} \leq t \leq 1, $$ for \(f\) and \(g\) in \(L(X)\). We define homotopy of loops, written \(f \sim g\), if there is a continuous function \(F:[0,1] \times[0,1] \rightarrow X\) such that \(F(s, 0)=F(s, 1)=x_{0}\) for every \(s\) and \(F(0, t)=f(t), F(1, t)=g(t)\) for every \(t\). Show that the set \(\pi(X)\) of equivalence classes (under homotopy) of loops is a group under the product \(\pi(f) \pi(g)=\pi(f g)\), where \(\pi: L(X) \rightarrow \pi(X)\) is the quotient map. Hint: If \(F\) is a homotopy between the loops \(f_{1}\) and \(f_{2}\), and \(G\) is a homotopy between the loops \(g_{1}\) and \(g_{2}\), set $$ \begin{array}{cll} H(s, t)=F(s, 2 t) & \text { for } & 0 \leq s \leq 1, & 0 \leq t \leq \frac{1}{2} \\ (s, t)=G(s, 2 t-1) & \text { for } & 0 \leq s \leq 1, & \frac{1}{2} \leq t \leq 1 \end{array} $$ and check that \(H\) is a homotopy between \(f_{1} g_{1}\) and \(f_{2} g_{2}\). The product in \(\pi(X)\) is therefore well-defined. If \(f \in L(X)\), define \(f^{-1}\) in \(L(X)\) by \(f^{-1}(t)=f(1-t)\) and check that \(f^{-1} f \sim e\), where \(e(t)=x_{0}\) for all \(t\). The relevant homotopy is $$ \begin{array}{lll} F(s, t)=f(2 s t) & \text { for } & 0 \leq s \leq 1, & 0 \leq t \leq \frac{1}{2} \\ F(s, t)=f(2 s(1-t)) & \text { for } & 0 \leq s \leq 1, & \frac{1}{2} \leq t \leq 1 \end{array} $$ Similarly \(f f^{-1} \sim e, f e \sim e f \sim f\), so that \(\pi(e)\) is the identity in \(\pi(X)\). Given \(f, g, h\) in \(L(X)\) we have $$ \begin{aligned} &f(g h)(t)= \begin{cases}h(4 t) & \text { for } 0 \leq t \leq \frac{1}{4} \\ g(4 t-1) & \text { for } \frac{1}{4} \leq t \leq \frac{1}{2} \\ f(2 t-1) & \text { for } & \frac{1}{2} \leq t \leq 1\end{cases} \\ &(f g) h(t)= \begin{cases}h(2 t) & \text { for } 0 \leq t \leq \frac{1}{2} \\ g(4 t-2) & \text { for } \frac{1}{2} \leq t \leq \frac{3}{4} \\ f(4 t-3) & \text { for } \frac{3}{4} \leq t \leq 1\end{cases} \end{aligned} $$ To show that \(f(g h) \sim(f g) h\), use the homotopy $$ F(s, t)= \begin{cases}h\left(4 t(1+s)^{-1}\right) & \text { for } 4 t-1 \leq s \\\ g(4 t-s-1) & \text { for } 4 t-2 \leq s \leq 4 t-1 \\ f\left((4 t-s-2)(2-s)^{-1}\right) & \text { for } s \leq 4 t-2\end{cases} $$

A set is called countable (or countably infinite) if it has the same cardinality (cf. E 1.1.6) as the set No natural numbers. Show that there is a well-ordered set \((X, \leq)\), which is itself uncountable, but which has the property that each segment min \(\\{x\\}\) is countable if \(x \in X\). Hint: Choose a well-ordered set \((Y, \leq)\) that is uncountable. The subset \(Z\) of elements \(z\) in \(Y\) such that the segment \(\min \\{z\\}\) is uncountable is either empty (and we are done) or else has a first element \(\Omega\) Set \(X=\min \\{\Omega\\}\). The ordinal number (corresponding to) \(\Omega\) is called the first uncountable ordinal.

Let \(Y\) be a dense subset of a topological space \((X, \tau)\). Show that \((Y \cap A)^{-}=A^{-}\)for every open subset \(A\) of \(X .\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.