Physics Exam > Physics Questions > If A ⊂ R2 is a countable set, then R2 \ A...
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If A ⊂ R2 is a countable set, then R2 \ A is
- a)Connected
- b)Countable
- c)Uncountable
- d)not connected
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
If A ⊂ R2 is a countable set, then R2 \ A isa)Connectedb)Countable...
Proof. Let x, y ∈ R
2 − A. Since there are only countably many points in A,
there are only countably many lines through x that hit A. Similarly, there
are only countably many lines through y that hit A. Therefore there are
at least two lines through x that do not hit A and one line through y that
does not hit A. (Actually there are uncountably many, but this is all we
need.) Then the line through y must intersect one of those two lines through
x in a point z. We can then form a path from x to y in R
2 − A by taking
the union of the line segment from x to z with the line segment from z to
y and appropriately parameterizing this geometric construction to form a
continuous function f : [0, 1] → R
2 with f(0) = x and f(1) = y
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If A ⊂ R2 is a countable set, then R2 \ A isa)Connectedb)Countable...
Is true, then B is true as well.
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If A ⊂ R2 is a countable set, then R2 \ A isa)Connectedb)Countablec)Uncountabled)not connectedCorrect answer is option 'B'. Can you explain this answer? for Physics 2025 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about If A ⊂ R2 is a countable set, then R2 \ A isa)Connectedb)Countablec)Uncountabled)not connectedCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Physics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If A ⊂ R2 is a countable set, then R2 \ A isa)Connectedb)Countablec)Uncountabled)not connectedCorrect answer is option 'B'. Can you explain this answer?.
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