MHB Find a countable set that is also open

  • Thread starter Thread starter seacoast123
  • Start date Start date
  • Tags Tags
    Set
Click For Summary
No countable subset of the real line can be open. The argument begins by assuming a countable open set C exists in the real numbers and selecting any point x from C. For x, there exists a delta such that the interval (x - delta, x + delta) is entirely contained within C. However, this interval is uncountable, leading to the conclusion that C cannot be countable. Thus, a countable open set in the real line cannot exist.
seacoast123
Messages
1
Reaction score
0
Find a countable set that is also open or prove that one cannot exist
 
Physics news on Phys.org
seacoast123 said:
Find a countable set that is also open or prove that one cannot exist
No countable subset of the real line is open. To prove it, assume $C$ is a countable open subset of $\mathbb R$ and $x$ be any point in $C$.

Then there exists $\delta>0$ such that $(x-\delta,x+\delta)\subseteq C$.

But $(x-\delta,x+\delta)$ is uncountable (why?).

Hence $C$ cannot be countable.
 

Thread 'A variant of the Monty Hall problem'

There is a nice little variation of the problem. The host says, after you have chosen the door, that you can change your guess, but to sweeten the deal, he says you can choose the two other doors, if you wish. This proposition is a no brainer, however before you are quick enough to accept it, the host opens one of the two doors and it is empty. In this version you really want to change your pick, but at the same time ask yourself is the host impartial and does that change anything. The host...
View full post »

Similar threads

I Countability of binary Strings
  • · Replies 18 ·
Replies
18
Views
2K
I The set of nonzero values of pmf is at most countable
  • · Replies 3 ·
Replies
3
Views
3K
MHB Proving that a subset of a countable set is countable
  • · Replies 1 ·
Replies
1
Views
3K
I Construction of sigma-algebras: a counterexample
  • · Replies 5 ·
Replies
5
Views
2K
MHB Proving Countability: Positive Rational Numbers & Union of Countable Sets
  • · Replies 11 ·
Replies
11
Views
3K
I Is an algorithm for a proof required to halt?
  • · Replies 3 ·
Replies
3
Views
2K
A Comparing infinite countable sets
  • · Replies 22 ·
Replies
22
Views
2K
I Countably Infinite Unions and the Real Numbers: Can They Really Be Uncountable?
  • · Replies 4 ·
Replies
4
Views
2K
I Proof of Countability of ℚ: Bijection from A to ℕ
  • · Replies 8 ·
Replies
8
Views
2K
I Countability of Finite Product Sets: Foundational Issues?
  • · Replies 2 ·
Replies
2
Views
1K