Uncountability of neighborhoods in R^n

[variety]

Quick question: Is it true that all neighborhoods in R^n form an uncountable set? It seems obvious to me that the answer is yes, but there is no proof in my analysis book and I can't think of one.

 

[Hurkyl]

Quick question: Is it true that all neighborhoods in R^n form an uncountable set? It seems obvious to me that the answer is yes, but there is no proof in my analysis book and I can't think of one.

Some bases for the topology on R^n are uncountable, and some bases are countably infinite. It depends on which one you have chosen.

 

[variety]

How about just a neighborhood of the real line centered at x with radius r>0.

 

[rochfor1]

Try proving the the set of all balls around any single point is uncountable, and think about it's relationship to your problem.

 

[morphism]

Wait, what exactly are you asking? Do you want to prove that the set {y : |x-y|<r} is uncountable? Or do you want to prove the collection of all such neighborhoods is uncountable?

 

[variety]

Yeah sorry I wasn't that clear. I just want to know if the set {x in R : |x-y|<r} is uncountable.

 

[adriank]

Well, that is just your open interval (y - r, y + r), which is indeed uncountable. (A bijection from the real numbers to this interval is f(x) = y + r \tanh(x).)