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November 27th, 2017, 03:31 PM   #1
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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?
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November 27th, 2017, 05:26 PM   #2
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Originally Posted by Joppy View Post
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?
The topological space defines which sets are open. Then a neighborhood is defined in terms of open sets. So the topological space is logically prior.

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Originally Posted by Joppy View Post
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 $p$.
What is a circle in a topological space? Might your book be referring to metric spaces, where there is a notion of distance?


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Can I assume that this 'disk' is analogous to the open set $U$ referred to in the above articles?
No, because a topological space has no notion of distance.

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.
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Last edited by Maschke; November 27th, 2017 at 05:30 PM.
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November 27th, 2017, 06:01 PM   #3
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Originally Posted by Maschke View Post
The topological space defines which sets are open. Then a neighborhood is defined in terms of open sets. So the topological space is logically prior.
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Originally Posted by Maschke View Post
What is a circle in a topological space? Might your book be referring to metric spaces, where there is a notion of distance?
Thanks.

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 cherry-picked):

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Let $p$ be a point in the plane. A neighborhood of $p$ is any circular disk without the boundary circle that contains $p$. Let $A$ be a subset of the plane. The point $p$ is called near the set $A$ if every neighborhood of $p$ contains a point of $A$
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November 27th, 2017, 06:04 PM   #4
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Originally Posted by Maschke View Post

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.
Right. This helps. I think I'll need to read a bit further for it to sink in.
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November 27th, 2017, 10:19 PM   #5
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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.
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November 27th, 2017, 10:39 PM   #6
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Is your book confusing on this point? Shouldn't be. Is this an obscure book of some sort?
Possibly.. I don't think it is supposed to be a rigorous treatment since it is an introduction, and no requirement of analysis is needed. Nevertheless I get confused when trying to find more information on the definitions presented.

It is stated in the preface that ideas from point-set topology is developed as needed, the focus being on combinatorial topology and applications.
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November 27th, 2017, 11:27 PM   #7
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Possibly.. I don't think it is supposed to be a rigorous treatment since it is an introduction, and no requirement of analysis is needed. Nevertheless I get confused when trying to find more information on the definitions presented.

It is stated in the preface that ideas from point-set topology is developed as needed, the focus being on combinatorial topology and applications.
Oh I see. Well they're being a little confusing but perhaps it won't matter in the context of the text.

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.
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Last edited by Maschke; November 27th, 2017 at 11:30 PM.
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