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

[Math Amateur]

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:

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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 \(\displaystyle [a,b]\) ... ... ?
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***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?---------------------------------------------------------------------------------------------------------------------------------

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

Peter

 

[Olinguito]

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 $[a,\,b]$.

 

[Math Amateur]

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