Uncountable family of disjoint closed sets

  • Thread starter sazanda
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  • #1
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Homework Statement


Determine whether the following statements are true or false
a) Every pairwise disjoint family of open subsets of ℝ is countable.
b) Every pairwise disjoint family of closed subsets of ℝ is countable.

Homework Equations


part (a) is true. we can find 1-1 correspondence with rational numbers

But part (b) I know it is false. I need a counter example. Could you help me with that?


The Attempt at a Solution

 

Answers and Replies

  • #2
Dick
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Homework Statement


Determine whether the following statements are true or false
a) Every pairwise disjoint family of open subsets of ℝ is countable.
b) Every pairwise disjoint family of closed subsets of ℝ is countable.

Homework Equations


part (a) is true. we can find 1-1 correspondence with rational numbers

But part (b) I know it is false. I need a counter example. Could you help me with that?


The Attempt at a Solution

You are probably thinking too hard. Think of sets consisting of a single element. Those are closed, yes?
 
  • #3
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You are probably thinking too hard. Think of sets consisting of a single element. Those are closed, yes?
Let me clarify myself.
let X be a collection of disjoint closed sets. Define X := { {x} such that x in ℝ }
{x}_1 is the one of the disjoint closed set.
{x}_2 is another disjoint closed set.
and so fourth
{x}_i is the another disjoint closed set
Since ℝ is uncountable X must be uncountable.

Is this what you mean?
 
  • #4
Deveno
Science Advisor
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the way you are listing the {x}_i, makes it look as if X is countable.

but in fact, |X| = |U(x in R){x}| = |R|, because we have a bijection from X to R:

{x}<---> x
 

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