Question
Download Solution PDFConsider β2 with the usual Euclidean metric. Let
π = {(π₯, π₯ sin \(\frac{1}{x}\)) ∈ β2 βΆ π₯ ∈ (0,1]} β {(0, π¦) ∈ β2 : −∞ < π¦ < ∞} and
π = {(π₯, sin \(\frac{1}{x}\)) ∈ β2 : π₯ ∈ (0,1]} β {(0, π¦) ∈ β2 : −∞ < π¦ < ∞}.
Consider the following statements:
π: π is a connected subset of β2 .
π: π is a connected subset of β2 .
Then
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFExplanation -
π = {(π₯, π₯ sin \(\frac{1}{x}\)) ∈ β2 βΆ π₯ ∈ (0,1]} β {(0, π¦) ∈ β2 : −∞ < π¦ < ∞}
{(0, π¦) ∈ β2 : −∞ < π¦ < ∞} represents the whole y - axis and π₯ sin \(\frac{1}{x}\) is oscillates between 0 and 1. Hence it is connected.
So Clearly X is also connected subset of β2 .
Hence Statement P is correct.
π = {(π₯, sin \(\frac{1}{x}\)) ∈ β2 : π₯ ∈ (0,1]} β {(0, π¦) ∈ β2 : −∞ < π¦ < ∞}.
{(0, π¦) ∈ β2 : −∞ < π¦ < ∞} represents the whole y - axis and sin \(\frac{1}{x}\) is oscillates between 0 and 1. Hence it is connected.
So Clearly Y is also connected subset of β2 .
Hence Statement Q is correct.
Hence option (1) is correct.