June 23rd, 2019, 11:12 PM  #1 
Member Joined: Apr 2017 From: India Posts: 73 Thanks: 0  Neighborhood of a point
Is a subset of a Neighborhood of a point is also a Neighborhood of the point? Justify This question is being asked as an exercise after explaining the definition of Neighborhood of a point. My deduction: At first go, it seems yes the statement is true. However, it says subset and not necessarily an interval. So, if I pick up any two elements randomly , then it will form a subset and it can happen that it doesn't contain any interval in which the point belongs. And hence, not every subset of a neighborhood of a point can be a neighborhood of the point. Is my reasoning correct? 
June 23rd, 2019, 11:15 PM  #2 
Senior Member Joined: Oct 2009 Posts: 863 Thanks: 328 
If you say no, not every subset of a neighborhood is a neighborhood, then you need to provide a counterexample. Only a counterexample is a valid reasoning.

June 23rd, 2019, 11:24 PM  #3 
Member Joined: Apr 2017 From: India Posts: 73 Thanks: 0 
Counterexample: (1,1) is a neighborhood of 0. However {0.5} being a singleton subset of this interval is not a neighborhood of the point. Is this reasoning correct? I meant, that is this counterexample valid? 
June 23rd, 2019, 11:32 PM  #4 
Senior Member Joined: Oct 2009 Posts: 863 Thanks: 328 
That works yes.


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