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Example 5: B(0, t) = {(X1, X2); |X1| < t, |x₂] < t} in (R², ||· ||∞) is an open ball. But {(X1X2, 0): |x₁| < t, |x₂| < t} is not open in (R³, ||· ||∞). So the projection map P: R³ R³. .P(X1, X2, X3) = (x1, x2, 0) is not an open map. Whereas Q : R³ → R². Q(X1, X2, X3) = (x1, x2) is an open map.

Example 5: B(0, t) = {(X1, X2); |X1| < t, |x₂] < t} in (R², ||· ||∞) is an open ball. But {(X1X2, 0): |x₁| < t, |x₂| < t} is not open in (R³, ||· ||∞). So the projection map P: R³ R³. .P(X1, X2, X3) = (x1, x2, 0) is not an open map. Whereas Q : R³ → R². Q(X1, X2, X3) = (x1, x2) is an open map.

Advanced Engineering Mathematics
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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Request explain the attached example

Request explain this example Example 5: B(0, t) = {(x1, x2); |X1| < t, |x₂| < t} in (R², ||· ||∞) is an open ball. But {(x1x2,0) |x₁|< t, |x₂| < t} is not open in (R³, ||· ||∞). So the projection map P: R³ R³. .P(X1, X2, X3) = (x1, x2, 0) is not an open map. Whereas Q : R³ → R². Q(X1, X2, X3) = (x1, x2) is an open map.
Transcribed Image Text:Request explain this example Example 5: B(0, t) = {(x1, x2); |X1| < t, |x₂| < t} in (R², ||· ||∞) is an open ball. But {(x1x2,0) |x₁|< t, |x₂| < t} is not open in (R³, ||· ||∞). So the projection map P: R³ R³. .P(X1, X2, X3) = (x1, x2, 0) is not an open map. Whereas Q : R³ → R². Q(X1, X2, X3) = (x1, x2) is an open map.
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