4. Show thatis open in R³.X := {(x, y, z) = R³ : x+y+z‡# 1}

Answered: 4. Show that is open in P3 X := {(x, y,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

4. Show that
is open in R³.
X := {(x, y, z) = R³ : x+y+z‡# 1}

Expert Solution

Step 1: To define a function:

One must solve the problem with the help of topology.

$L e t space u s space d e f i n e space a space f u n c t i o n space f colon space straight real numbers cubed rightwards arrow straight real numbers space d e f i n e d space b y space f left parenthesis x comma y comma z right parenthesis equals x plus y plus z minus 1. A g a i n space l e t space u s space d e f i n e space t h e space s e t space X equals left curly bracket left parenthesis x comma y comma z right parenthesis element of straight real numbers cubed vertical line f left parenthesis x comma y comma z right parenthesis not equal to 0 right curly bracket H e n c e space t h e space r a n g e space s e t space o f space f space r e s t r i c t e d space o n space X space i s space t h e space u n i o n space o f space t w o space o p e n s space s e t s left parenthesis negative infinity comma 0 right parenthesis union left parenthesis 0 comma infinity right parenthesis.$