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Exercise 6.45. Let E be a nonempty open set in R. For each x € E, let Ix = (ax,bx), where ax = inf {a ER: (a,x) CE} and b = sup {b R: (x, b) CE}. (i) Prove that E = Ux€E Ix. TEE (ii) Prove that if x € E, y € E, and IxIy ‡ Ø, then Ix = Iy. (iii) Prove that the collection U = {I: x € E} is countable.

Exercise 6.45. Let E be a nonempty open set in R. For each x € E, let Ix = (ax,bx), where ax = inf {a ER: (a,x) CE} and b = sup {b R: (x, b) CE}. (i) Prove that E = Ux€E Ix. TEE (ii) Prove that if x € E, y € E, and IxIy ‡ Ø, then Ix = Iy. (iii) Prove that the collection U = {I: x € E} is countable.

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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Exercise 6.45. Let E be a nonempty open set in R. For each x E E, let Ix = (ax, bx), where ax = inf {a ER: (a,x) CE} and b sup {bER: (x, b) CE}. (i) Prove that E=UTEE IT. (ii) Prove that if x E E, y E E, and IxIy ‡ Ø, then I = Iy. (iii) Prove that the collection U = {I: x € E} is countable. =
Transcribed Image Text:Exercise 6.45. Let E be a nonempty open set in R. For each x E E, let Ix = (ax, bx), where ax = inf {a ER: (a,x) CE} and b sup {bER: (x, b) CE}. (i) Prove that E=UTEE IT. (ii) Prove that if x E E, y E E, and IxIy ‡ Ø, then I = Iy. (iii) Prove that the collection U = {I: x € E} is countable. =
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