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Compact Topological Spaces ... Stromberg, Example 3.34 (c) ... ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,910
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 an aspect of Example 3.34 (c) on page 102 ... ...


Examples 3.34 (plus some relevant definitions ...) reads as follows:




Stromberg - Example 3.34 (c) ...  .png




In Example 3.34 (c) above from Stromberg we read the following:

" ... ... Let \(\displaystyle \mathscr{I}\) be the collection of all open intervals \(\displaystyle I\) such that \(\displaystyle I \subset U\) for some \(\displaystyle U\) in \(\displaystyle \mathscr{U}\). Check that \(\displaystyle \mathscr{I}\) is a cover of \(\displaystyle [a,b]\) ... ... "




My question is as follows:

How would we go about (rigorously) checking that \(\displaystyle \mathscr{I}\) is a cover of \(\displaystyle [a,b]\) ... ... indeed how would we rigorously demonstrate that \(\displaystyle \mathscr{I}\) is a cover of \(\displaystyle [a,b]\) ... ... ?



------------------------------------------------------------------------------------------------------------------------------

***EDIT***

My thoughts ... after reflecting ...


\(\displaystyle \mathscr{U}\) is an open cover (family of open subsets) of \(\displaystyle [a, b]\) ... ...

Each set \(\displaystyle U \subset \mathscr{U} \) is a countable set of pairwise disjoint open intervals ... ... (Theorem 3.18)

Therefore if \(\displaystyle \mathscr{I}\) equals the collection of all open intervals \(\displaystyle I\) such that \(\displaystyle I \subset U\) ...

... then \(\displaystyle \mathscr{I} \) is a family of open intervals such that \(\displaystyle [a, b] \subset \bigcup \mathscr{I}\) ...

Now apply Heine-Borel Theorem ...


Is that correct?


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Hope someone can help ...

Peter



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


The above text from Stromberg mentions the Heine_Borel Theorem ... so I am proving the text of the (statement of ...) the theorem ... ... as follows:



Stromberg -  Statement of Heine-Borel Theorem 1.66 ... .png



"My thoughts ... ... " above include a reference to Theorem 3.18 ... the text of the statement of Theorem 3.18 is as follows:



Stromberg - 1 - Statement of Theorem 3.18 ... PART 1 ... .png
Stromberg - 2 - Statement of Theorem 3.18 ... PART 2  ... .png






Hope that helps ...

Peter
 
Last edited:

Olinguito

Well-known member
Apr 22, 2018
251
Let $x\in[a,\,b]$. Then $x\in U$ for some $U\in\mathscr U$. As $U$ is open, there is an open interval $I_x$ such that $x\in I_x\subseteq U$. So $I_x\in\mathscr I$ and $[a,\,b]\subseteq\bigcup_xI_x$, i.e. $\mathscr I$ is a cover for $[a,\,b]$.
 

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,910
Let $x\in[a,\,b]$. Then $x\in U$ for some $U\in\mathscr U$. As $U$ is open, there is an open interval $I_x$ such that $x\in I_x\subseteq U$. So $I_x\in\mathscr I$ and $[a,\,b]\subseteq\bigcup_xI_x$, i.e. $\mathscr I$ is a cover for $[a,\,b]$.



Thanks for the help Olinguito

Peter