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\section*{Are there gaps in the rationals? Version \%}
There is a sense in which there is no empty space in the rational numbers: no matter how close together two rationals are, there are infinitely many other rationals between them. In this
sense, there are no gaps in the rationals.
However, there is a sense in which there \emph{is} empty space in the rational numbers. Let \[
A=\left\{ r\in\mathbb{Q}:r>0 \text{ and } r^{2}\leq2\right\} \hspace{1em}\text{and}\hspace{1em}B=\left\{ s\in\mathbb{Q}:s>0 \text{ and } s^{2}\geq2\right\}. \]
Clearly, any element of $A$ is less than
any element of $B$. Nonetheless, there is no element $q\in\mathbb{Q}$
such that $q$ is between $A$ and $B$. Thus it is not always
possible to find a rational number between any two \emph{sets }of rationals. In this sense, there are gaps in the rationals.
To make matters geometrically
explicit, consider the graph of $y=2-x^{2}$ using only rational numbers. Notice that there's no $x\in\mathbb{Q}$ that makes $2-x^{2}=0$. Thus, the graph never crosses the $x$-axis, despite the fact that the graph is above the $x$-axis at $x=0$ and below
the $x$-axis at $x=3$. This demonstrates
the existence of a more devious type of gap in the rationals, and
the failure of $\mathbb{Q}$ to model our intuition about the physical
world.
\newpage
\emph{
Show that every infinite subset of $[0, 1] \times [0, 1]$ has a
limit point.
}
Let $K = [0, 1] \times [0, 1]$ be the unit square and let $S
\subseteq K$. We proceed by proving the contrapositive: If $S$ has
no limit points, then $S$ is finite.
Suppose that $S$ has no limit points. Then for each $p \in K$, there
exists some ball around $p$ not containing any point of $S$, except
perhaps possibly for $p$ itself. Finitely many of these balls cover
$K$, and this cover also covers $S$. Thus, $S$ is finite because it can be covered by finitely many balls. Taking the
contrapositive, if $S$ is an infinite subset of $[0, 1] \times [0,
1]$, then $S$ has a limit point.
\newpage
\section*{Are there gaps in the rationals? Version \#}
There is a sense in which there is no empty space in the rational numbers: no matter how close together two rationals are, there are infinitely many other rationals between them. In this
sense, there are no gaps in the rationals.
However, there is a sense in which there \emph{is} empty space in the rational numbers: while it is possible to find a rational number
between any two other rational numbers, it is \emph{not} always
possible to find a rational number between any two \emph{sets }of rationals. For example, let \[
A=\left\{ r\in\mathbb{Q}:r>0 \text{ and } r^{2}\leq2\right\} \hspace{1em}\text{and}\hspace{1em}B=\left\{ s\in\mathbb{Q}:s>0 \text{ and } s^{2}\geq2\right\}. \]
Clearly, any element of $A$ is less than
any element of $B$. Nonetheless, there is no element $q\in\mathbb{Q}$
such that $q$ is between $A$ and $B$. In this sense, there is a gap in the rationals between sets $A$ and $B$.
To make matters geometrically
explicit, consider the graph of $y=2-x^{2}$ using only rational numbers.
At $x=0$, the graph is above the $x$-axis, and at $x=3$ it is below
the $x$-axis. However, the graph never actually crosses the $x$-axis, because
there's no $x\in\mathbb{Q}$ that makes $2-x^{2}=0$. This demonstrates
the existence of a more devious type of gap in the rationals, and
the failure of $\mathbb{Q}$ to model our intuition about the physical
world.
\end{document}