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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 9213
Then ... 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 \(\displaystyle B_\epsilon (x) \subset U\) ... ... "
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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
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 9213
Then ... 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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