/*! 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} Q 64. Prove that if S is a closed subs... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Q 64. (page 932)

Prove that if S is a closed subset of R2or R3, then Se is an open set. This is Theorem 12.12

Short Answer

Expert verified

It is proved that if Sis a closed subset of R2or R3, then Seis an open set .

Step by step solution

01

Step 1. Given information. 

We have given closed subset S ofR2orR3.

02

 Prove the given statement.

The objective is to prove the theorem which states that, "If S is a closed subset of R2orR3then Seis an open set."

Assume S is a closed subset of R2orR3.

The definition of a closed set is given as, 鈥淎 subset of R2or R3is said to be closed, if its complement is an open set."

Thus, from the above definition, S can be termed to be closed, if its complement Se is open. Combining the two statements, it is proved that if S is a closed subset of R2or R3, then Seis an open set .

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!

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

Use Theorem 12.33 to find the indicated derivatives in Exercises 27鈥�30. Express your answers as functions of two variables.

鈭�z鈭�rwhenz=(x2+xy)ey,x=rcos胃andy=rsin胃

Describe the meanings of each of the following mathematical expressions :

鈭�A

Use Theorem 12.32 to find the indicated derivatives in Exercises 21鈥�26. Express your answers as functions of a single variable.

dydtwheny=rsin胃,r=t3,and胃=t

In Exercises, find the maximum and minimum of the function f subject to the given constraint. In each case explain why the maximum and minimum must both exist.

f(x,y)=xywhenx2+y2=16

Extrema:Findthelocalmaxima,localminima,andsaddlepointsofthegivenfunctionsf(x,y)=x4+y4+4xy
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

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.