Carothers' Definitions: Neighborhoods, Open Sets, and Open Balls

In summary, N. L. Carothers defines an open ball, neighborhood, and open set as follows: an open ball is a set that contains an open ball about every point in the set; a neighborhood is a set that contains an open ball about every point in the set, but not necessarily an open ball about every point in the set; an open set is a set that contains an open ball about every point in the set.
  • #1
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The Definition of a Neighborhood and the Definition of an Open Set ... Carothers, Chapters 3 & 4 ...

I am reading N. L. Carothers' book: "Real Analysis". ... ...

I am focused on Chapter 3: Metrics and Norms and Chapter 4: Open Sets and Closed Sets ... ...

I need help with an aspect of Carothers' definitions of open balls, neighborhoods and open sets ...Now ... on page 45 Carothers defines an open ball as follows:
View attachment 9213Then ... on page 46 Carothers defines a neighborhood as follows:
View attachment 9214
And then ... on page 51 Carothers defines an open set as follows:
View attachment 9215
Now my question is as follows:

When Carothers re-words his definition of an open set he says the following:

" ... ... In other words, \(\displaystyle U\) is an open set if, given \(\displaystyle x \in U\), there is some \(\displaystyle \epsilon \gt 0\) such that \(\displaystyle B_\epsilon (x) \subset U \) ... ... "
... BUT ... in order to stay exactly true to his definition of neighborhood shouldn't Carothers write something like ..." ... ... In other words, \(\displaystyle U\) is an open set if, for each \(\displaystyle x \in U\), \(\displaystyle U\) contains a neighborhood \(\displaystyle N\) of \(\displaystyle x\) such that \(\displaystyle N\) contains an open ball \(\displaystyle B_\epsilon (x)\) ... ..."Can someone lease explain how, given his definition of neighborhood he arrives at the statement ...

" ... ... In other words, \(\displaystyle U\) is an open set if, given \(\displaystyle x \in U\), there is some \(\displaystyle \epsilon \gt 0\) such that \(\displaystyle B_\epsilon (x) \subset U\) ... ... "

=========================================================================================

Reflection ... maybe we can regard \(\displaystyle B_\epsilon (x)\) as a neighborhood contained in U since \(\displaystyle B_{ \frac{ \epsilon }{ 2} }(x)\) \(\displaystyle \subset\) \(\displaystyle B_\epsilon (x)\) ... is that correct?But then why doesn't Carothers just define a neighborhood of \(\displaystyle x\) as an open ball about \(\displaystyle x\) ... rather than a set containing an open ball about \(\displaystyle x\)?=========================================================================================

Hope someone can clarify ...

Peter
 

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  • #2
Re: The Definition of a Neighborhood and the Definition of an Open Set ... Carothers, Chapters 3 & 4

Peter said:
... BUT ... in order to stay exactly true to his definition of neighborhood shouldn't Carothers write something like ..." ... ... In other words, \(\displaystyle U\) is an open set if, for each \(\displaystyle x \in U\), \(\displaystyle U\) contains a neighborhood \(\displaystyle N\) of \(\displaystyle x\) such that \(\displaystyle N\) contains an open ball \(\displaystyle B_\epsilon (x)\) ... ..."
If \(\displaystyle U\) contains a neighborhood \(\displaystyle N\) of \(\displaystyle x\) such that \(\displaystyle N\) contains an open ball \(\displaystyle B_\epsilon (x)\) then $x\in B_\epsilon (x) \subseteq N \subseteq U$, so it is certainly true that $B_\epsilon (x) \subseteq U$. Conversely, $B_\epsilon (x)$ is a neighbourhood of $x$, so if $B_\epsilon (x) \subseteq U$ then we can take $N = B_\epsilon (x)$. It will then be true that "\(\displaystyle U\) contains a neighborhood \(\displaystyle N\) of \(\displaystyle x\) such that \(\displaystyle N\) contains an open ball \(\displaystyle B_\epsilon (x)\)".

Notice that an open set containing $x$ is a neighbourhood of $x$. But a neighbourhood of $x$ need not be an open set contining $x$ (because a neighbourhood does not have to be open).
 
  • #3
Re: The Definition of a Neighborhood and the Definition of an Open Set ... Carothers, Chapters 3 & 4

Opalg said:
If \(\displaystyle U\) contains a neighborhood \(\displaystyle N\) of \(\displaystyle x\) such that \(\displaystyle N\) contains an open ball \(\displaystyle B_\epsilon (x)\) then $x\in B_\epsilon (x) \subseteq N \subseteq U$, so it is certainly true that $B_\epsilon (x) \subseteq U$. Conversely, $B_\epsilon (x)$ is a neighbourhood of $x$, so if $B_\epsilon (x) \subseteq U$ then we can take $N = B_\epsilon (x)$. It will then be true that "\(\displaystyle U\) contains a neighborhood \(\displaystyle N\) of \(\displaystyle x\) such that \(\displaystyle N\) contains an open ball \(\displaystyle B_\epsilon (x)\)".

Notice that an open set containing $x$ is a neighbourhood of $x$. But a neighbourhood of $x$ need not be an open set contining $x$ (because a neighbourhood does not have to be open).
Thanks for the help Opalg ...

Peter
 

1. What is the definition of a neighborhood in mathematics?

A neighborhood in mathematics is a set of points that are close to a given point. It can also be described as a region around a point that contains other points.

2. How are open sets defined in mathematics?

An open set in mathematics is a set of points that does not contain its boundary points. It is a set that is completely contained within its own interior.

3. What is the difference between a neighborhood and an open set?

The main difference between a neighborhood and an open set is that a neighborhood is defined based on a specific point, while an open set is defined based on its boundary points. Additionally, a neighborhood can be open or closed, while an open set is always open.

4. How are open balls defined in mathematics?

An open ball in mathematics is a set of points that are within a given distance from a specific point. It is similar to a neighborhood, but it is defined based on distance rather than a specific point.

5. Why are Carothers' definitions important in mathematics?

Carothers' definitions of neighborhoods, open sets, and open balls are important in mathematics because they provide a formal and precise way of defining concepts that are crucial in many areas of mathematics, such as topology and analysis. These definitions help to establish a common language and framework for understanding and solving mathematical problems.

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