Definition 40.
A point is simply an element of the real numbers \(\R\text{.}\)
Section 2.1 Point Sets
Definition 41.
A point set is a nonempty subset of the real numbers.
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.
Axiom 42. Point Set Axiom 1.
If \(p\) is a point, then there is a point less than \(p\) and a point greater than \(p\text{.}\)
Axiom 43. Point Set Axiom 2.
If \(p\ne q\) are two points, then there is a point between them.
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.
Example 44.
Definition 45.
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*}
If \(N\) is an open interval and \(x\in N\text{,}\) we will sometimes say that \(N\) is a neighborhood of \(x\text{.}\)
We will often be interested in examining small neighborhoods around particular points.
Problem 47.
Let \(a,b\in\R\text{.}\)
- 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.
Definition 48.
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*}
In either case, \(a\) and \(b\) are called the endpoints of the interval.
Example 49.
Give some examples of point sets that are not composed of intervals.
Definition 50.
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 \(x\in M\setminus \set{p}\text{.}\)
Problem 51.
Show that if \(M\) is an open interval and \(p\in M\text{,}\) then \(p\) is an accumulation point of \(M\text{.}\)
Problem 52.
Show that if \(M\) is a closed interval and \(p\notin M\text{,}\) then \(p\) is not an accumulation point of \(M\text{.}\)
Problem 53.
Determine whether the endpoints of an open interval \(M\) are accumulation points of \(M\text{.}\)
It is worth exploring exactly how many points it is possible or impossible for \(M\) to have. The next two problems start in investigating that.
Problem 54.
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.
Problem 55.
Consider \(\Z\) as a point set in \(\R\text{.}\) Show that it has no accumulation points.
Problem 56.
Given that \(H\) and \(K\) are point sets, show either of the following (your choice):
- If \(p\) is an accumulation point of \(H\cap K\text{,}\) then \(p\) is an accumulation point of both \(H\) and \(K\text{.}\)
- 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 \(K\text{.}\)
Problem 57.
If \(M\) is the set of all reciprocals of elements of \(\N\text{,}\) then 0 is an accumulation point of \(M\text{.}\)
We will now start to turn to connecting the concepts of point sets to more familiar ones from calculus, beginning with sequences.
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