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

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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.
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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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• Carothers - Defn of an Open Set ... Page 51 ... .png
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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).

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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