Math Amateur
Gold Member
MHB
- 3,920
- 48
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, $$U$$ is an open set if, given $$x \in U$$, there is some $$\epsilon \gt 0$$ such that $$B_\epsilon (x) \subset U $$ ... ... "
... BUT ... in order to stay exactly true to his definition of neighborhood shouldn't Carothers write something like ..." ... ... In other words, $$U$$ is an open set if, for each $$x \in U$$, $$U$$ contains a neighborhood $$N$$ of $$x$$ such that $$N$$ contains an open ball $$B_\epsilon (x)$$ ... ..."Can someone lease explain how, given his definition of neighborhood he arrives at the statement ...
" ... ... In other words, $$U$$ is an open set if, given $$x \in U$$, there is some $$\epsilon \gt 0$$ such that $$B_\epsilon (x) \subset U$$ ... ... "
=========================================================================================
Reflection ... maybe we can regard $$B_\epsilon (x)$$ as a neighborhood contained in U since $$B_{ \frac{ \epsilon }{ 2} }(x)$$ $$\subset$$ $$B_\epsilon (x)$$ ... is that correct?But then why doesn't Carothers just define a neighborhood of $$x$$ as an open ball about $$x$$ ... rather than a set containing an open ball about $$x$$?=========================================================================================
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 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, $$U$$ is an open set if, given $$x \in U$$, there is some $$\epsilon \gt 0$$ such that $$B_\epsilon (x) \subset U $$ ... ... "
... BUT ... in order to stay exactly true to his definition of neighborhood shouldn't Carothers write something like ..." ... ... In other words, $$U$$ is an open set if, for each $$x \in U$$, $$U$$ contains a neighborhood $$N$$ of $$x$$ such that $$N$$ contains an open ball $$B_\epsilon (x)$$ ... ..."Can someone lease explain how, given his definition of neighborhood he arrives at the statement ...
" ... ... In other words, $$U$$ is an open set if, given $$x \in U$$, there is some $$\epsilon \gt 0$$ such that $$B_\epsilon (x) \subset U$$ ... ... "
=========================================================================================
Reflection ... maybe we can regard $$B_\epsilon (x)$$ as a neighborhood contained in U since $$B_{ \frac{ \epsilon }{ 2} }(x)$$ $$\subset$$ $$B_\epsilon (x)$$ ... is that correct?But then why doesn't Carothers just define a neighborhood of $$x$$ as an open ball about $$x$$ ... rather than a set containing an open ball about $$x$$?=========================================================================================
Hope someone can clarify ...
Peter
Attachments
Last edited: