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

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In summary, Peter was asking how to prove that a given set (mathscr I) is a cover of a given set (a, b). The Heine-Borel theorem is a theorem that can be used to prove this.
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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 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

Attachments

• Stromberg - Example 3.34 (c) ... .png
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• Stromberg - Statement of Heine-Borel Theorem 1.66 ... .png
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• Stromberg - 1 - Statement of Theorem 3.18 ... PART 1 ... .png
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• Stromberg - 2 - Statement of Theorem 3.18 ... PART 2 ... .png
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Last edited:
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 $[a,\,b]$.

Olinguito said:
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

1. What is a compact topological space?

A compact topological space is a mathematical concept that describes a space in which every open cover has a finite subcover. In simpler terms, this means that every collection of open sets that covers the space can be reduced to a finite number of sets that still cover the space.

2. How is compactness different from connectedness?

Compactness and connectedness are two different properties of a topological space. Compactness refers to the ability to cover the space with a finite number of sets, while connectedness refers to the ability to move continuously from one point to another in the space without leaving it. A space can be compact without being connected, and vice versa.

3. What is the significance of Example 3.34 (c) in Stromberg's book?

Example 3.34 (c) in Stromberg's book is significant because it provides an example of a compact topological space that is not Hausdorff. This goes against the common belief that all compact spaces are also Hausdorff, and highlights the importance of understanding the subtle differences between topological properties.

4. How are compact topological spaces used in mathematics?

Compact topological spaces are used in many areas of mathematics, including analysis, geometry, and topology. They are particularly useful in proving theorems and solving problems related to continuity, convergence, and compactness. They also have applications in physics, computer science, and other fields.

5. Can every topological space be made compact?

No, not every topological space can be made compact. In fact, compactness is a very specific property that not all spaces possess. For example, infinite spaces or spaces with infinitely many points cannot be made compact. Additionally, some spaces may have certain topological properties that prevent them from being compact.

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