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\title{Basic Topology of $\mathbb{R}$ (continued)}
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\section{Compact Sets}     
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\begin{definition}[compact set] A set $K\subseteq\mathbb{R}$ is \textbf{compact} if every sequence in $K$ has a subsequence that converges to an element of $K$.
\end{definition}
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\begin{example} A closed interval $[a,b]$ is an example of a compact set.
\end{example}
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The idea of compactness extends some of the properties that finite sets enjoy to a larger class of possibly infinite sets.    
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\begin{definition}[bounded set] The set $E\subseteq\mathbb{R}$ is \textbf{bounded} if there exists  $M>0$ such that
\[
|a|\leq M\quad\mbox{for all}\quad a\in E
\]
\end{definition}
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\begin{theorem}[Heine-Borel Theorem] A set $K\subseteq\mathbb{R}$ is compact if and only if it is closed and bounded.
\end{theorem}
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Now we generalize the Nested Interval property to compact sets:
\begin{theorem} If $K_1\supseteq K_2\supseteq K_3\cdots$ is a nested sequence of nonempty compact sets, then
\[
\bigcap_{n=1}^\infty K_n\quad\neq\quad\emptyset
\]
\end{theorem}
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Next we consider an important alternative characterization of compact sets in terms of open sets.
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\begin{definition}[open cover] An \textbf{open cover} for a set $E\subseteq\mathbb{R}$ is a collection of open sets
\[
\{O_{\alpha}\;:\; \alpha\in I\}
\]
(where $I$ is some index set) with the property that
\[
E\subseteq\bigcup_{\alpha\in I} O_{\alpha}
\]
\end{definition}
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\begin{definition}[finite subcover] 
If
\[
\{O_{\alpha}\;:\; \alpha\in I\}
\]
is an open cover of $E$ a \textbf{finite subcover} $S=S_1,S_2,\ldots,S_n$ is a finite subcollection of open sets in $O$ that still covers $E$:
\[
E\subseteq S_1\cup S_2\cup \cdots\cup S_n
\]
\end{definition}
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\begin{theorem} For any set $E\subseteq\mathbb{R}$, the following statements are equivalent (that is to say, if any one is true, they are all true):
\begin{itemize}
\item $E$ is compact
\item $E$ is closed and bounded
\item $E$ Every open cover of $E$ has a finite subcover
\end{itemize}
\end{theorem}
\section{Connected Sets}
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\begin{definition}[separated] Two nonempty sets $A,B\subseteq\mathbb{R}$ are called \textbf{separated} if
\[
\overline{A}\cap B\;=\;\emptyset\;=\;A\cap\overline{B}
\]
\end{definition}
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\begin{definition}[disconnected] A set $E\subseteq\mathbb{R}$ is said to be \textbf{disconnected} if it can be written 
in the form
\[
E=A\cup B
\]
where $A$ and $B$ are nonempty separated sets.  As set which is not disconnected is said to be \textbf{connected}.
\end{definition}
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\begin{definition}[totally disconnected] A set $E$ is said to be  \textbf{totally disconnected} if, for any two distinct elements $a,b\in E$, there exist separated sets $A$ and $B$ with $a\in A$, $b\in B$, and $E=A\cup B$.
\end{definition}

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