One of the interesting things about choosing to look at the whole real number line is that we have two additional useful axioms. They seem nearly trivial, but are very important.
Note that these are closely related to the last few problems—but we are starting from scratch here, and you should treat these as the primary axioms we need for now.
If instead of \(\R\) we chose as our “background set” some subset \(C\) of \(\R\text{,}\) these axioms might not hold. Let \(C = [0,1]\cup [2,3]\text{,}\) and illustrate the failure of each of Axiom 42 and Axiom 43.
We say that a point set \(O\) is an open interval if there are points \(a\ne b\) such that \(O\) is the point set consisting of all points between \(a\) and \(b\text{.}\) That is,
\begin{equation*}
O = \setof{x}{a \lt x \lt b} = (a,b).
\end{equation*}
For a positive real number \(\epsilon\) and any \(x\in \R\text{,}\) the \(\epsilon\) neighborhood of \(x\) is the neighborhood \(N_\epsilon(x) = (x-\epsilon,x+\epsilon)\text{.}\)
Using the notion of an open interval, formulate a definition for what it would mean for \(a\approx b\text{.}\) Then illustrate your concept with two different pairs of real numbers.
Again using the notion of an open interval, formulate a definition for what it would mean for \(a = b\text{.}\) Explain why your definition makes sense.
We say that a point set \(I\) is a closed interval if there are points \(a\ne b\) such that \(I\) is the point set consisting of \(a,b\text{,}\) and all points between \(a\) and \(b\text{.}\) That is,
\begin{equation*}
I = \setof{x}{a \le x\le b} = [a,b].
\end{equation*}
If \(M\) is a point set, we say that \(p\) is an accumulation point of \(M\) if every open interval containing \(p\) also contains a point of \(M\) different from \(p\text{.}\)
To word this in set notation, \(p\) is an accumulation point of \(M\) if, for every open interval \(O\) with \(p\in O\text{,}\) there is \(x\in O\) such that \(x\in M\setminus \set{p}\text{.}\)
Show that if \(M\) is a point set having an accumulation point, then \(M\) contains (at least) two points. Determine whether \(M\) must contain at least three points.
If \(p\) is an accumulation point of \(H\cup K\text{,}\) then \(p\) is an accumulation point of \(H\) or \(p\) is an accumulation point of \(K\text{.}\)
