August 25th, 2017, 07:28 PM  #1 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 211 Thanks: 2  Neighbourhood of a point Neighbourhood : A subset N of R is a neighbourhood of a point p if $\displaystyle \exists$ $\displaystyle \epsilon > 0 $ such that $\displaystyle (x \epsilon, x+\epsilon) \subset N $ Q : A set which is a neighbourhood of all its points except $n$ points $\displaystyle (n \geq 1)$. Can this set be an Interval ? A : The set $\displaystyle (0,1) \cup $ {$1,2,...,n$} is a neighbourhood of all its points except $n$ points $\displaystyle 1,2,3...n$ If $\displaystyle n=1$, the set can be an interval. Ex : $\displaystyle (2,3]$ If $\displaystyle n=2$, the set can be an interval. Ex : $\displaystyle [2,3]$ If $\displaystyle n>2$, the set cannot be an interval. Can someone explain me the set ($\displaystyle (0,1) \cup $ {$1,2,...,n$}) defined here ? 
August 25th, 2017, 09:45 PM  #2 
Senior Member Joined: Aug 2012 Posts: 1,709 Thanks: 458  You mean just explain what that set is? Well, it's the open unit interval (0,1) unioned with the finite set of points {1, 2, 3, ..., n}. That's pretty self explanatory. It's the set of points that are either in the open unit interval or else they're one of the positive integers between 1 and n inclusive.

August 25th, 2017, 09:59 PM  #3  
Senior Member Joined: Nov 2015 From: hyderabad Posts: 211 Thanks: 2  Quote:
Why it is not an interval if $\displaystyle n \geq 2$  
August 25th, 2017, 10:58 PM  #4 
Senior Member Joined: Aug 2012 Posts: 1,709 Thanks: 458  
August 25th, 2017, 11:20 PM  #5 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 211 Thanks: 2  
August 26th, 2017, 09:04 AM  #6  
Senior Member Joined: Aug 2012 Posts: 1,709 Thanks: 458  Quote:
In other words I need to see a formal proof that 2 is not a neighborhood of N. What does containing infinitely many points have to do with it? Which values of $\epsilon$? etc. "$N$ is not a neighborhood of any of it's points."  Isn't N a neighborhood of, say, 1/2? Last edited by Maschke; August 26th, 2017 at 09:52 AM.  
August 26th, 2017, 07:48 PM  #7  
Senior Member Joined: Nov 2015 From: hyderabad Posts: 211 Thanks: 2  Quote:
Let us take $\displaystyle \epsilon = 1/2=0.5 >0 $ and the open Interval $\displaystyle (20.5,2+0.5) = (1.5,2.5) \not\subset N$ Hence, $N$ is not a neighbourhood of its points. The Infinitely many points refer to the Rational and Irrational numbers, together the Reals. I'm not much proficient in writing the proofs, please let me know if any corrections to be made.  
August 26th, 2017, 08:08 PM  #8  
Senior Member Joined: Aug 2012 Posts: 1,709 Thanks: 458  Quote:
Quote:
Quote:
I'm only talking about the unit interval. Let's just deal with that case first. There's some confusion. I'm taking N to be the unit interval plus some isolated points, and I'm asking if it's a neighborhood of $\frac{1}{2}$. In your proof that N is not a neighborhood of 2, wouldn't you have to prove that EVERY epsilon fails? Do you need to review your quantifier negation? The negation of $\exists x P(x)$ is $\forall x \neg P(x)$. Last edited by Maschke; August 26th, 2017 at 08:15 PM.  
August 26th, 2017, 08:14 PM  #9 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 211 Thanks: 2 
If we take the natural number $1$, in case of $x=2$, the definition will work with the open interval $\displaystyle (1,3) \subset N$, since you are telling that we do only consider Natural numbers in this case.

August 26th, 2017, 08:16 PM  #10  
Senior Member Joined: Aug 2012 Posts: 1,709 Thanks: 458  Quote:
Do you see why the open unit interval $(0,1)$ is a neighborhood of $\frac{1}{2}$? We need to understand this before looking at the rest of it.  

Tags 
neighbourhood, point 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Nearest distance from point to point on line  abcdefgh123  Algebra  2  January 4th, 2014 04:08 PM 
the measure of 0neighbourhood for functions with large grad  medvedev_ag  Real Analysis  0  June 5th, 2013 02:38 AM 
Drawing a line from point a,b .. coords point b unkown  Vibonacci  Algebra  5  September 17th, 2012 01:28 AM 
calculating point of line given starting point, slope  mathsiseverything  Algebra  1  March 4th, 2008 07:41 AM 
the measure of 0neighbourhood for functions with large grad  medvedev_ag  Calculus  0  December 31st, 1969 04:00 PM 