November 27th, 2017, 04:31 PM  #1 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,749 Thanks: 613 Math Focus: Yet to find out.  Neighbourhood of point
In this Wiki article, a neighbourhood of a point in a topological space is defined, where a topological space is a set of points with each point having a set of neighbourhoods. I don't get it.. which comes first? The definition of neighbourhood or the space? Also, the book I'm reading gives the following definition of a neighbourhood: A neighbourhood of $p$ is any circular disk without the boundary circle that contains $p$. Can I assume that this 'disk' is analogous to the open set $U$ referred to in the above articles? 
November 27th, 2017, 06:26 PM  #2  
Senior Member Joined: Aug 2012 Posts: 2,135 Thanks: 621  Quote:
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Now in a metric space, we can define open discs (aka open balls) in terms of the distance function; then define open sets in terms of open balls; and then prove that the collection of open sets satisfies the rules for being a topology. In other words there's a concrete approach using the idea of a distance; and an abstract approach in which a topological space has a collection of open sets that are essentially arbitrary. Last edited by Maschke; November 27th, 2017 at 06:30 PM.  
November 27th, 2017, 07:01 PM  #3  
Senior Member Joined: Feb 2016 From: Australia Posts: 1,749 Thanks: 613 Math Focus: Yet to find out.  Quote:
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That's why I am confused. There is no explicit mention of metric space, although the Euclidean norm is defined on the previous page so I guess that's a giveaway.. Skipping ahead, a topological space is defined in terms of neighbourhoods. But as you suggest, how do we have a circle (used to define neighbourhood) if there is no distance. That definition is actually the first in the book and is actually a definition on nearness (sorry, shouldn't have cherrypicked): Quote:
 
November 27th, 2017, 07:04 PM  #4  
Senior Member Joined: Feb 2016 From: Australia Posts: 1,749 Thanks: 613 Math Focus: Yet to find out.  Quote:
 
November 27th, 2017, 11:19 PM  #5 
Senior Member Joined: Aug 2012 Posts: 2,135 Thanks: 621 
If they define the Euclidean norm that's going to give you a metric space, since you can use the norm to define a metric. That is, if $\lvert x \rvert$ is the norm of $x$, then $\lvert x  y \rvert$ is a metric (needs proof). You can't define a circle in a general topological space unless it also happens to be a metric space. Is your book confusing on this point? Shouldn't be. Is this an obscure book of some sort? Reason I ask is, "A neighborhood of p is any circular disk without the boundary circle that contains p" is true, but an unusual way to define a neighborhood.

November 27th, 2017, 11:39 PM  #6  
Senior Member Joined: Feb 2016 From: Australia Posts: 1,749 Thanks: 613 Math Focus: Yet to find out.  Quote:
It is stated in the preface that ideas from pointset topology is developed as needed, the focus being on combinatorial topology and applications.  
November 28th, 2017, 12:27 AM  #7  
Senior Member Joined: Aug 2012 Posts: 2,135 Thanks: 621  Quote:
You can think of a neighborhood as a little region around a point. The region should either be an open set or should contain an open set. Different authors use one or the other definition  that a neighborhood must either be open or merely contain an open set. According to Wiki, combinatorial topology is an old name for algebraic topology. Now if you're studying algebraic topology it might be helpful to know some general topology and some abstract algebra. Or perhaps your book will define everything you need. I can't say. Last edited by Maschke; November 28th, 2017 at 12:30 AM.  

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