Problem 3.33. Let X = {0,1} and Y = [0,1] with their natural orders. Put the dictionary order on Y x X. Put the order topology on Y × X. Answer the following: (1) Place in order (0,0), (1,0), (0, 1), (1, 1). (2) Is [0, 1] × {0} an open set? (3) Is (0, 1] × {0} a closed set? Je the oint in the order betaueen (O 1) and (1.0)2

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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Solve 3.33 all the parts in detail 

Problem 3.33. Let X =
{0,1} and Y = [0, 1] with their natural orders. Put the dictionary
order on Y x X. Put the order topology on Y × X. Answer the following:
(1) Place in order (0,0), (1,0), (0, 1), (1, 1).
(2) Is [0, 1] × {0} an open set?
(3) Is (0, 1] × {0} a closed set?
(4) Is there any point in the order between (0, 1) and (1,0)?
(5) Does this order have both a least and a greatest element?
(6) What points of Y x X are limit points?
Transcribed Image Text:Problem 3.33. Let X = {0,1} and Y = [0, 1] with their natural orders. Put the dictionary order on Y x X. Put the order topology on Y × X. Answer the following: (1) Place in order (0,0), (1,0), (0, 1), (1, 1). (2) Is [0, 1] × {0} an open set? (3) Is (0, 1] × {0} a closed set? (4) Is there any point in the order between (0, 1) and (1,0)? (5) Does this order have both a least and a greatest element? (6) What points of Y x X are limit points?
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