Uncountability of neighborhoods in R^n

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Discussion Overview

The discussion revolves around the question of whether all neighborhoods in R^n form an uncountable set. Participants explore this concept within the context of topology and analysis, examining specific cases and definitions related to neighborhoods and open intervals.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • Some participants propose that all neighborhoods in R^n are uncountable, suggesting that this seems obvious but lacks a proof in standard analysis texts.
  • Others argue that the countability of neighborhoods depends on the chosen basis for the topology on R^n, noting that some bases are uncountable while others are countably infinite.
  • A participant questions whether the focus is on proving that the set {y : |x-y|
  • Another participant clarifies that the specific set {x in R : |x-y|
  • A suggestion is made to prove that the set of all balls around any single point is uncountable, indicating a potential relationship to the original question.

Areas of Agreement / Disagreement

Participants express differing views on the countability of neighborhoods, with some asserting uncountability while others highlight the dependency on the chosen topology. The discussion remains unresolved regarding the broader question of all neighborhoods in R^n.

Contextual Notes

The discussion does not resolve the assumptions regarding the definitions of neighborhoods and bases for topology, nor does it clarify the implications of different topological choices on the countability of neighborhoods.

variety
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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.
 
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variety said:
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.
 
How about just a neighborhood of the real line centered at x with radius r>0.
 
Try proving the the set of all balls around any single point is uncountable, and think about it's relationship to your problem.
 
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?
 
Yeah sorry I wasn't that clear. I just want to know if the set {x in R : |x-y|<r} is uncountable.
 
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).)
 

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