Heine-Borel Theorem/Real Line/Closed and Bounded Set

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Theorem

Let $F$ be a closed and bounded real set.

Let $C$ be a set of open real sets.

Let $C$ be a cover of $F$.


Then there is a finite subset of $C$ that covers $F$.


Proof

We are given that $C$ is a set of open real sets that covers $F$.

In other words, $C$ is an open cover of $F$.

We need to show that there is a finite subset of $C$ that covers $F$.

In other words, we need to show that $C$ has a finite subcover.


Let $F_o$ be the complement of $F$ in $\R$.

By the definition of closed real set, $F_o$ is open as $F$ is closed.


Step 1: $C^*$ is an Open Cover of $\left[{a \,.\,.\, b}\right]$

It is demonstrated that $C^*$ is an open cover of $\left[{a \,.\,.\, b}\right]$.


Since $F$ is bounded, $F$ is contained in a closed and bounded interval $\left[{a \,.\,.\, b}\right]$ where $a, b \in \R$.

Define $C^* = C \cup \left\{ {F_o}\right\}$.

Like $C$, $C^*$ is a set of open real sets as $F_o$ is open.


$C^*$ covers $F \cup F_o$ as $C$ covers $F$ and $\left\{ {F_o} \right\}$ covers $F_o$.

$F_o \cup F$ equals $\R$ as $F_o$ is the complement of $F$ in $\R$.

So $C^*$ covers $\R$.

Furthermore, $C^*$ is an open cover of $\left[{a \,.\,.\, b}\right]$ as $\left[{a \,.\,.\, b}\right]$ is a subset of $\R$.


Step 2: $C^*$ has a Finite Subcover

It is demonstrated that $C^*$ has a finite subcover $C^*_f$.


$C^*$ is an open cover of the closed and bounded interval $\left[{a \,.\,.\, b}\right]$.

Therefore, by Open Cover of Closed and Bounded Real Interval has Finite Subcover, $C^*$ has a finite subcover $C^*_f$.


Step 3: $C$ has a Finite Subcover

It is demonstrated that $C$ has a finite subcover $C_f$.


Note that $F_o$ is the only element in $C^*$ that is not an element of $C$.

Therefore, $F_o$ is the only possible element in $C^*_f$ that is not an element of $C$ as $C^*_f$ is a subset of $C^*$.

This means that $C^*_f$ \ $\left\{ {F_o} \right\}$ is a subset of $C$.


Define $C_f$ = $C^*_f$ \ $\left\{ {F_o} \right\}$.

According to the reasoning above, $C_f$ is a subset of $C$.

Also, $C_f$ is finite as $C^*_f$ is finite.

What remains is to show that $C_f$ covers $F$.


We have:

\(\displaystyle F \subseteq \left[ {a \,.\,.\, b} \right]\) \(\implies\) \(\displaystyle F \subseteq \left[ {a \,.\,.\, b} \right] \subseteq \bigcup_{O \mathop \in C^*_f} O\) as $C^*_f$ is a cover of $\left[ {a \,.\,.\, b} \right]$
\(\displaystyle \) \(\implies\) \(\displaystyle F \subseteq \bigcup_{O \mathop \in C^*_f} O\)
\(\displaystyle \) \(\implies\) \(\displaystyle F \cap \complement F_o \subseteq (\bigcup_{O \mathop \in C^*_f} O) \cap \complement F_o\) Set Intersection Preserves Subsets
\(\displaystyle \) \(=\) \(\displaystyle F \cap \complement \complement F \subseteq (\bigcup_{O \mathop \in C^*_f} O) \cap \complement F_o\) as $F_o = \complement F$, the complement of $F$ in $\R$
\(\displaystyle \) \(=\) \(\displaystyle F \cap F \subseteq (\bigcup_{O \mathop \in C^*_f} O) \cap \complement F_o\) Relative Complement of Relative Complement
\(\displaystyle \) \(=\) \(\displaystyle F \subseteq (\bigcup_{O \mathop \in C^*_f} O) \cap \complement F_o\) Intersection is Idempotent

Furthermore, as $F \subseteq \left[{a \,.\,.\, b}\right]$:

\(\displaystyle F\) \(\subseteq\) \(\displaystyle \left({\bigcup_{O \mathop \in C^*_f} O}\right) \cap \complement F_o\)
\(\displaystyle \) \(\subseteq\) \(\displaystyle \left({\bigcup_{O \mathop \in C^*_f \cup \left\{ {F_o}\right\} } O}\right) \cap \complement F_o\) by $\displaystyle \left({\bigcup_{O \mathop \in C^*_f} O}\right) \subseteq \left({\bigcup_{O \mathop \in C^*_f \cup \left\{ {F_o}\right\} } O}\right)$ and Set Intersection Preserves Subsets
\(\displaystyle \) \(=\) \(\displaystyle \left({\bigcup_{O \mathop \in C^*_f \cup \left\{ {F_o}\right\} } O}\right) \setminus F_o\) as set intersection with complement equals set difference
\(\displaystyle \) \(=\) \(\displaystyle \left({\bigcup_{O \mathop \in \left({C^*_f \setminus \left\{ {F_o}\right\} \cup \left\{ {F_o}\right\} }\right)} O}\right) \setminus F_o\) by Set Difference Union Second Set is Union
\(\displaystyle \) \(=\) \(\displaystyle \left({\left({\bigcup_{O \mathop \in \left({C^*_f \setminus \left\{ {F_o}\right\} }\right)} O}\right) \cup F_o}\right) \setminus F_o\)
\(\displaystyle \) \(=\) \(\displaystyle \left({\bigcup_{O \mathop \in \left({C^*_f \setminus \left\{ {F_o}\right\} }\right)} O}\right) \setminus F_o\) Set Difference with Union is Set Difference
\(\displaystyle \) \(\subseteq\) \(\displaystyle \bigcup_{O \mathop \in \left({C^*_f \setminus \left\{ {F_o}\right\} }\right)} O\) Set Difference is Subset
\(\displaystyle \) \(=\) \(\displaystyle \left({\bigcup_{O \mathop \in C_f} O}\right)\) Definition of $C_f$

Thus, $C_f$ covers $F$.

This finishes the proof.

$\blacksquare$


Source of Name

This entry was named for Heinrich Eduard Heine and Émile Borel.


Sources