Baire Category Theorem .... Stromberg, Theorem 3.55 .... ....

  • #1
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I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ...

I am focused on Chapter 3: Limits and Continuity ... ...

I need help in order to fully understand the proof of Theorem 3.55 on page 110 ... ...


Theorem 3.55 and its proof read as follows:



View attachment 9165






At the start of the second paragraph of the above proof by Stromberg we read the following:

" ... ...Since \(\displaystyle A_1^{ - \ \circ } = \emptyset\), we can choose \(\displaystyle x_1\) in the open set \(\displaystyle V\) \ \(\displaystyle A_1^{ - }\) and then we can choose \(\displaystyle 0 \lt r_1 \lt 1\) such that \(\displaystyle B_{ r_1 } ( x_1 )^{ - } \subset V\) \ \(\displaystyle A_1^{ - }\) [ check that \(\displaystyle B_r (x)^{ - } \subset B_{ 2r } (x) \) ] ... ...





My questions are as follows:


Question 1

Can someone explain and demonstrate why/how it is that \(\displaystyle A_1^{ - \ \circ } = \emptyset\) means that we can choose \(\displaystyle x_1\) in the open set \(\displaystyle V\) \ \(\displaystyle A_1^{ - }\) ... how are we (rigorously) sure this is true ... ?



Question 2

How/why can we choose \(\displaystyle 0 \lt r_1 \lt 1\) such that \(\displaystyle B_{ r_1 } ( x_1 )^{ - } \subset V\) \ \(\displaystyle A_1^{ - }\) ... ?

... and why are we checking that \(\displaystyle B_r (x)^{ - } \subset B_{ 2r } (x)\) ... ... ?



*** EDIT ***

My thoughts on Question 2 ...

Since \(\displaystyle V\) \ \(\displaystyle A_1^{ - }\) is open ... \(\displaystyle \exists \ r_1\) such that \(\displaystyle B_{ r_1 } ( x_1 ) \subset V\) \ \(\displaystyle A_1^{ - }\) ...

... BUT ... how do we formally and rigorously show that ...

... we can choose an \(\displaystyle r_1\) such that the closure of \(\displaystyle B_{ r_1 } ( x_1 )\) is a subset of \(\displaystyle V\) \ \(\displaystyle A_1^{ - }\) ...

... (intuitively I think we just choose \(\displaystyle r_1\) somewhat smaller yet ...)

... and further why is Stromberg talking about \(\displaystyle r_1\) between \(\displaystyle 0\) and \(\displaystyle 1\) ...?






Help will be much appreciated ...

Peter



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

The definitions of nowhere dense, first and second category and residual are relevant ... so I am providing Stromberg's definitions ... as follows:



View attachment 9166




Stromberg's terminology and notation associated with the basic notions of topological spaces are relevant to the above post ... so I am providing the text of the same ... as follows:




View attachment 9167





Hope that helps ...

Peter
 

Attachments

  • Stromberg - Theorem 3.55 ... Baire Category Theorem ... .png
    Stromberg - Theorem 3.55 ... Baire Category Theorem ... .png
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  • Stromberg -  Defn 3.53 ... Nowhere Dense ...First and Second Category ... .png
    Stromberg - Defn 3.53 ... Nowhere Dense ...First and Second Category ... .png
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  • Stromberg -  Defn 3.11  ... Terminology for Topological Spaces ... .png
    Stromberg - Defn 3.11 ... Terminology for Topological Spaces ... .png
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Last edited:

Answers and Replies

  • #2
Opalg
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Question 1

Can someone explain and demonstrate why/how it is that \(\displaystyle A_1^{ - \ \circ } = \emptyset\) means that we can choose \(\displaystyle x_1\) in the open set \(\displaystyle V\) \ \(\displaystyle A_1^{ - }\) ... how are we (rigorously) sure this is true ... ?
If $V\setminus A_1^-$ is empty then $V\subseteq A_1^{-\mathrm o}$, contradicting the fact that $A_1^{-\mathrm o}$ is empty. Therefore $V\setminus A_1^-$ is not empty and so it contains some point, which we can choose as $x_1$.


Question 2

How/why can we choose \(\displaystyle 0 \lt r_1 \lt 1\) such that \(\displaystyle B_{ r_1 } ( x_1 )^{ - } \subset V\) \ \(\displaystyle A_1^{ - }\) ... ?

... and why are we checking that \(\displaystyle B_r (x)^{ - } \subset B_{ 2r } (x)\) ... ... ?



*** EDIT ***

My thoughts on Question 2 ...

Since \(\displaystyle V\) \ \(\displaystyle A_1^{ - }\) is open ... \(\displaystyle \exists \ r_1\) such that \(\displaystyle B_{ r_1 } ( x_1 ) \subset V\) \ \(\displaystyle A_1^{ - }\) ...

... BUT ... how do we formally and rigorously show that ...

... we can choose an \(\displaystyle r_1\) such that the closure of \(\displaystyle B_{ r_1 } ( x_1 )\) is a subset of \(\displaystyle V\) \ \(\displaystyle A_1^{ - }\) ...

... (intuitively I think we just choose \(\displaystyle r_1\) somewhat smaller yet ...)

... and further why is Stromberg talking about \(\displaystyle r_1\) between \(\displaystyle 0\) and \(\displaystyle 1\) ...?
Your thoughts on Question 2 are quite correct. If you can choose a number, say $s$ such that $B_{ s } ( x_1 ) \subset V\setminus A_1^{ - }$, then let $r_1 = \frac12s$. Then $s=2r_1$, so that $B_{ 2r_1 } ( x_1 ) \subset V\setminus A_1^{ - }$ and hence the closure of $B_{ r_1 } ( x_1 )$ is contained in $V\setminus A_1^{ - }$.

Given $r_1$ with that property, any smaller (positive) value of $r_1$ will have the same property. So you can always assume that $0<r_1<1$. Stromberg's reason for wanting that is that this is the first step of an inductive construction in which he wants to ensure that $r_n\to0$ as $n\to\infty$. The easiest way to do that is to require that $r_n<1/n$.
 
  • #3
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If $V\setminus A_1^-$ is empty then $V\subseteq A_1^{-\mathrm o}$, contradicting the fact that $A_1^{-\mathrm o}$ is empty. Therefore $V\setminus A_1^-$ is not empty and so it contains some point, which we can choose as $x_1$.



Your thoughts on Question 2 are quite correct. If you can choose a number, say $s$ such that $B_{ s } ( x_1 ) \subset V\setminus A_1^{ - }$, then let $r_1 = \frac12s$. Then $s=2r_1$, so that $B_{ 2r_1 } ( x_1 ) \subset V\setminus A_1^{ - }$ and hence the closure of $B_{ r_1 } ( x_1 )$ is contained in $V\setminus A_1^{ - }$.

Given $r_1$ with that property, any smaller (positive) value of $r_1$ will have the same property. So you can always assume that $0<r_1<1$. Stromberg's reason for wanting that is that this is the first step of an inductive construction in which he wants to ensure that $r_n\to0$ as $n\to\infty$. The easiest way to do that is to require that $r_n<1/n$.



Thanks Opalg ...

... very much appreciate your help...

Peter
 
  • #4
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Thanks Opalg ...

... very much appreciate your help...

Peter



Hi Opalg ...

Thanks again for your help ...

Just a further point of clarification ...

You write:

" ... ... Then $s=2r_1$, so that $B_{ 2r_1 } ( x_1 ) \subset V\setminus A_1^{ - }$ and hence the closure of $B_{ r_1 } ( x_1 )$ is contained in $V\setminus A_1^{ - }$. ... ... "


So ... as I understand it, you are arguing that because $B_{ 2r_1 } ( x_1 ) \subset V\setminus A_1^{ - }$ ...

... ... that therefore ... the closure of $B_{ r_1 } ( x_1 )$ is contained in $V\setminus A_1^{ - }$. ...

... that is that $B_{ r_1 } ( x_1 )^{ - } \subset V\setminus A_1^{ - } $ ... ...

But why ... rigorously ... is this true ... (... it certainly seems plausible ... but rigorously ...? )



Hope that you can help further ...

Peter
 
Last edited:
  • #5
Opalg
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... as I understand it, you are arguing that because $B_{ 2r_1 } ( x_1 ) \subset V\setminus A_1^{ - }$ ...

... ... that therefore ... the closure of $B_{ r_1 } ( x_1 )$ is contained in $V\setminus A_1^{ - }$. ...

... that is that $B_{ r_1 } ( x_1 )^{ - } \subset V\setminus A_1^{ - } $ ... ...

But why ... rigorously ... is this true ... (... it certainly seems plausible ... but rigorously ...? )
The result that you need is that if $x$ is an element in a metric space $(X,d)$, and $r>0$, then $B_r(x)^- \subseteq B_{2r}(x)$.

To see that, notice that if $y\in B_r(x)^-$ then there must be a point $z\in B_r(x)$ with $d(z,y)<r$. By the triangle inequality, $d(y,x) \leqslant d(y,z) + d(z,x) < r + r = 2r$. Therefore $y\in B_{2r}(x).$
 
  • #6
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The result that you need is that if $x$ is an element in a metric space $(X,d)$, and $r>0$, then $B_r(x)^- \subseteq B_{2r}(x)$.

To see that, notice that if $y\in B_r(x)^-$ then there must be a point $z\in B_r(x)$ with $d(z,y)<r$. By the triangle inequality, $d(y,x) \leqslant d(y,z) + d(z,x) < r + r = 2r$. Therefore $y\in B_{2r}(x).$




Thanks for the help Opalg ...

But ... just a point of clarification ...

You write:

" ... ... notice that if $y\in B_r(x)^-$ then there must be a point $z\in B_r(x)$ with $d(z,y)<r$ ... ... "


I can see what you write must be true (by definition) when \(\displaystyle y\) is a limit point of \(\displaystyle B_r(x)^-\) ... by why is this true when \(\displaystyle y\) is an interior point of \(\displaystyle B_r(x)^-\) ... given we are dealing with a general metric space ...



Hope you can help further ...

Peter
 
  • #7
Opalg
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You write:

" ... ... notice that if $y\in B_r(x)^-$ then there must be a point $z\in B_r(x)$ with $d(z,y)<r$ ... ... "


I can see what you write must be true (by definition) when \(\displaystyle y\) is a limit point of \(\displaystyle B_r(x)^-\) ... by why is this true when \(\displaystyle y\) is an interior point of \(\displaystyle B_r(x)^-\) ... given we are dealing with a general metric space ...
If \(\displaystyle y\in B_r(x)^-\) is not a limit point, then \(\displaystyle y\in B_r(x)\), which is obviously contained in \(\displaystyle B_{2r}(x)\).
 
  • #8
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If \(\displaystyle y\in B_r(x)^-\) is not a limit point, then \(\displaystyle y\in B_r(x)\), which is obviously contained in \(\displaystyle B_{2r}(x)\).




Thanks for all your help, Opalg ...

It is much appreciated ...

Peter
 

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