MHB 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:

View attachment 9122

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

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

My question is as follows:

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

***EDIT***

My thoughts ... after reflecting ...$$\mathscr{U}$$ is an open cover (family of open subsets) of $$[a, b]$$ ... ...

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

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

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

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:
View attachment 9123
"My thoughts ... ... " above include a reference to Theorem 3.18 ... the text of the statement of Theorem 3.18 is as follows:
View attachment 9124
View attachment 9125
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