Counting, uncounting, and dust, part six

Let’s quickly recap of some of the interesting things we have seen so far.

• The real numbers are uncountable, and in fact every interval of the real numbers is uncountable. In other words, there are more real numbers between 0 and 1 than there are integers from negative infinity to positive infinity. 
• However, this alone does not really give us a sense of just how much bigger uncountable is than countable. For instance, we can see that the sequence of fractional numbers \( \{ 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots \} \) is also contained within the interval \( (0, 1) \). It is clear, though, that this sequence has the cardinality of \( \mathbb{N} \). It is thus not really clear how it is so remarkable that there are more real numbers in \( (0, 1) \) than there are natural numbers anywhere at all.
• The Lebesgue measure of an isolated point is zero. Consequently, the Lebesgue measure of a countable number of isolated points is also zero. In particular, the Lebesgue measure of \( \mathbb{N} \), the natural numbers, is zero.
• But what if we take an uncountable number of isolated points? Wait, is that even possible?
• And what if we take “non-isolated” points, whatever that means?

Well, let’s begin by defining what it means to be an “isolated” point. We “know” intuitively that the points of \( \mathbb{N} \) are “isolated” when considered as a subset of the real numbers \( \mathbb{R} \). On the other hand, a set like \( \mathbb{R} \) itself is “dense”, which is the opposite of isolated. What does this mean? Formally, we define the notion of “denseness” in terms of a set (call it \( X \) ) and a subset of that set (call it \( A \) ). In this case, we say the subset \( A \) is dense in \( X \) if every point of \( X \) is either a point in \( A \) itself or arbitrarily close to a point in \( A \). What this intuitively means is that there is no getting away from points of \( A \) when we are inside \( X \); no matter where we go in \( X \) and no matter how tightly we try to close ourselves away, there will always be a point from \( A \) near us. 

It’s pretty clear that \( \mathbb{N} \) doesn’t fit this definition, because the points are all separate from each and equally spaced out over the real numbers. In this case, we say \( \mathbb{N} \) is nowhere dense in \( \mathbb{R} \). Surprisingly, denseness is true of the rational numbers! \( \mathbb{Q} \) is dense in \( \mathbb{R} \), even though it has “as many” elements as \( \mathbb{N} \). In other words, even though \( \mathbb{Q} \) and \( \mathbb{N} \) are both countable, their spatial arrangements (or more formally, their topological structures) are different, giving them different properties. We thus see that a countable set can be either dense in \( \mathbb{R} \) or nowhere dense in \( \mathbb{R} \).

It’s worth noting two features about nowhere dense sets:

• By one definition, a subset \( A \) is nowhere dense in a set \( X \) if the closure of \( A \) in \( X \) has an empty interior. As a reminder:
• The closure of \( A \) in \( X \) is the set of all those points in \( X \) that are limits of sequences of points taken only from \( A \). It should be immediately clear that the closure of \( A \) always includes \( A \), but may often include lots of other points too. Thus, the closure of \( \mathbb{Q} \) in \( \mathbb{R} \) is \( \mathbb{R} \) itself, which is clearly much larger than \( \mathbb{Q} \). In some intuitive sense, the closure of a set is a more “natural”, “self-contained” version of the set.
• The interior of \( A \) is the set of all those points strictly, properly within \( A \) itself; i.e., for every point in the interior of \( A \), it should be possible to establish some sort of exclusive zone around it that contains points only within \( A \) itself. So, for instance, the point  \( \frac{1}{2} \) is an interior point of the interval  \( [0, 1] \), but neither  \( 0 \)  nor  \( 1 \) are. Another example: the interior of \( \mathbb{Q} \) is the empty set (because you can always find some irrational number that is really close to every rational number, no matter how much you decide to “zoom in”).
• The complement in  \( X \) of a nowhere dense subset  \( A \) is a dense, open set.

[Strictly speaking, \( X \) cannot be any regular set; it must be a topological space, a kind of set in which it is possible to talk about ideas such as being connected, being continuous, being “near” or “far” in a loose sense.]

So to summarize, we have four different things here:

• the cardinality of a set, according to which the natural numbers \( \mathbb{N} \) and the rational numbers \( \mathbb{Q} \) are countable but the real numbers \( \mathbb{R} \) are uncountable;
• the counting measure of a set, according to which \( \mathbb{N} \), \( \mathbb{Q} \), and \( \mathbb{R} \) are all infinite
• the Lebesgue measure of a set, according to which both \( \mathbb{N} \) and \( \mathbb{Q} \) are sets of measure zero, \( \mathbb{R} \) has infinite Lebesgue measure, and any closed interval \( [a,b] \) has the finite measure \( b-a \).
• the denseness of a set in another set, according to which \( \mathbb{N} \) is nowhere dense in \( \mathbb{R} \), but \( \mathbb{Q} \) is dense in \( \mathbb{R} \).

Now, it would be natural to think that an uncountable set has to be dense, simply because there are so many more points in it (uncountably more, in fact!) than in a countable set. As it turns out, that’s not the case. This is where the Cantor set and other similar structures, sometimes whimsically called the “fat” Cantor sets (more formally, the Smith-Volterra-Cantor sets), come in.