Consider the closed rectangle R = [a,b] × [c,d] in ℝ2 (under the Euclidean metric). i. Explain why every horizontal line of the form [a,b] × {y} is compact. ii. Given any open cover Oλ of R, conclude that each line of the form [a,b] × {y} is covered by a finite subfamily of the Oλ's. iii. Using ii, show that R is compact. Hint: In iii, it may help to look at, for each horizontal line, the union of the finite subcover. How "thick" or "thin" can this union be? Note: Using the same argument as that for ℝ, it follows from this problem that a subset of ℝ2 is compact if and only if it is closed and bounded.
Consider the closed rectangle R = [a,b] × [c,d] in ℝ2 (under the Euclidean metric). i. Explain why every horizontal line of the form [a,b] × {y} is compact. ii. Given any open cover Oλ of R, conclude that each line of the form [a,b] × {y} is covered by a finite subfamily of the Oλ's. iii. Using ii, show that R is compact. Hint: In iii, it may help to look at, for each horizontal line, the union of the finite subcover. How "thick" or "thin" can this union be? Note: Using the same argument as that for ℝ, it follows from this problem that a subset of ℝ2 is compact if and only if it is closed and bounded.
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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Consider the closed rectangle R = [a,b] × [c,d] in ℝ2 (under the Euclidean metric).
i. Explain why every horizontal line of the form [a,b] × {y} is compact.
ii. Given any open cover Oλ of R, conclude that each line of the form [a,b] × {y} is covered by a finite subfamily of the Oλ's.
iii. Using ii, show that R is compact.
Hint: In iii, it may help to look at, for each horizontal line, the union of the finite subcover. How "thick" or "thin" can this union be?
Note: Using the same argument as that for ℝ, it follows from this problem that a subset of ℝ2 is compact if and only if it is closed and bounded.
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