The Affinely Extended Real Numbers

There are a great many topological spaces in the realm of mathematics, many of them quite exotic. Others are very similar to the systems we are familiar with. One of these is \overline{\mathbb{R}}, the affinely extended real numbers

The Definition of the Space

The idea of the extended real numbers is to add a pair of endpoints onto the real number line, thereby closing off the numbers between -\infty at one end and +\infty at the other. By simply applying the standard definitions of terms like “open set”, “limit point”, and so on, we will find that many nice results fall out more easily and naturally than in the ordinary real numbers. The downside, of course, is that arithmetic breaks down completely. While we can define some operations on \pm\infty, expressions like +\infty+(-\infty) must remain undefined. But such is the price we pay for being analysts rather than algebraists.

On \overline{\mathbb{R}} we define what is called the ‘order topology’. Specifically, we define our open intervals to be intervals of the form (a,+\infty], [-\infty,b), and (a,b), where a and b can be any point on the extended real line, including \pm\infty. Just as in the ordinary real numbers, we define the open sets as those consisting of some union of open intervals. This is slightly different than the ordinary real line, where open intervals are those of the form (a,+\infty), (-\infty,b), and (a,b), and where \pm\infty are not considered points in the space.

Note that in order to work properly with the extended real numbers, we will have to use the general topological definitions of terms like ‘open set’, ‘neighborhood’, ‘continuous’, and so on. Most of the real-number-specific definitions no longer work when \pm\infty is taken into account. For example, anything involving \delta-neighborhoods and \epsilon-neighborhoods is no longer usable in \overline{\mathbb{R}}: how would one construct an \epsilon-neighborhood of radius 2 around the point +\infty? In general point-set topology, the idea of a neighborhood is much less strict, and no notion of distance is required. A neighborhood of x is simply any superset of an open set containing x. For example, (0,6), (1,4), and [2,5) are all neighborhoods of 3 in both the ordinary reals and \overline{\mathbb{R}}, since the first two are themselves open sets containing 3 while the third is a superset of (2,5), which contains 3. However, [3,4) is not a neighborhood of 3 because no open set within [3,4) contains 3. Using this definition of ‘neighborhood’, we can now describe ideas like that of a limit of a sequence: (a_n) has a limit of (or converges to) L if every neighborhood of L contains all but finitely many a_n.

Because the definition of open sets in \overline{\mathbb{R}} is so similar to that of \mathbb{R}, it shouldn’t be too surprising that the types of sets that are open and closed remain generally similar. The only clopen sets (sets which are both open and closed) are the empty set and \overline{\mathbb{R}} itself. In this space, the set \mathbb{R} of real numbers is open, since it’s just an open interval of the form (a,b) where a = -\infty and b = +\infty. However, it is not closed because its complement \{-\infty,+\infty\} is not open. It is not that difficult to show that -\infty and +\infty are limit points of \mathbb{R}, so the closure of \mathbb{R} is \overline{\mathbb{R}}. This makes intuitive sense: we make \mathbb{R} closed by adding on the two endpoints at infinity.

Other examples of similar behavior can be found as well. In the ordinary real numbers \mathbb{N} is a closed set, but in \overline{\mathbb{R}} this is not quite the case. Note that the only open intervals containing +\infty are those of the form (a,+\infty] for some a. Each of these must contain all elements of the natural numbers greater than a, and therefore +\infty is a limit point of \mathbb{N}. In fact, the closure of \mathbb{N} is just \mathbb{N} \cup \{+\infty\}. On the other hand, the closure of \mathbb{Q} is all of \overline{\mathbb{R}} because the rationals are both dense and can be arbitrarily large.

Basic Properties of the Space

The most significant property of the extended reals is that they are bounded. Specifically, |r| \le +\infty for any r \in \overline{\mathbb{R}}. While this may seem obvious, it has a variety of consequences for the topology. Intuitively, because the reals are now bounded above, it seems that all closed sets should be compact. This turns out to be true, and follows from the proof below that \overline{\mathbb{R}} is a compact set.

Consider an arbitrary open cover O of \overline{\mathbb{R}}. Because of how we defined our space, the only open intervals containing +\infty are those of the form (a,+\infty], and therefore every open set containing +\infty has (a,+\infty] as a subset for some real number a. We can pick any such set S containing (a,+\infty] from the open cover, and similarly choose a set T (which may or may not be equal to S) containing [-\infty,b) for some b. We have used only two sets which must be in O, and all we have left to cover is the interval [a,b]. It is possible that this is empty (if b \le a), in which case we are done: \{S,T\} is a finite subcover. Otherwise, we know by the topology of the ordinary reals that [a,b] is compact, and therefore can be covered by a finite subcover of O. We can simply take that finite subcover and add on the two additional sets S and T to end up with a slightly larger (but clearly still finite) subcover of \overline{\mathbb{R}}.

It can be shown that any closed subset of a compact set is compact, so in fact every closed set in \overline{\mathbb{R}} is compact. This makes the Heine-Borel theorem much simpler to deal with in \overline{\mathbb{R}}. Because every closed set is both bounded and compact, the statements “K is compact” and “K is closed and bounded” are equivalent not only to each other but also to the even simpler statement “K is closed”.

From the equivalence of closedness and compactness, it can be shown that in \overline{\mathbb{R}} every monotone sequence converges, that every sequence has a convergent subsequence, and that every infinite set has a limit point. These are not just a nice simplification; they shed light on what is happening in the ordinary real numbers. If adding on endpoints of \pm\infty to the real line is enough to make every monotone sequence converge, this tells us that the only way a monotone sequence can diverge in \mathbb{R} is if it grows without bound, approaching where +\infty or -\infty would be, if we included those points in the space. Similarly, the only way a sequence in \mathbb{R} can lack a convergent subsequence is if all of its subsequences grow towards \pm\infty. These results could be demonstrated in \mathbb{R} itself, but they are much easier to prove and intuitively understand when we use \overline{\mathbb{R}}.

It is also worth mentioning that the Axiom of Completeness and the Nested Interval Property hold in \overline{\mathbb{R}}. On the other hand, the Archimedean Property is clearly false, since there is no natural number greater than +\infty. We can have AoC and NIP without the Archimedean Property because the proof of AP from AoC/NIP uses basic arithmetic, which cannot be relied upon when dealing with \pm\infty.

Sequences

We have shown that every sequence has a convergent subsequence, and that every monotone sequence converges. It should not be surprising, then, to see that many simple sequences in \overline{\mathbb{R}} converge towards \pm\infty despite being divergent in the ordinary reals.

For example, consider (x_n) = (x!). This grows rapidly and diverges in the ordinary reals, but in \overline{\mathbb{R}} every neighborhood of +\infty contains all but finitely many of these x_n, so we can say that the sequence of factorial numbers has a limit of +\infty.

Note that because we are dealing with sequences in \overline{\mathbb{R}}, we can include \pm\infty as points in sequences. From an analytic perspective this is not especially interesting, but it bears mentioning. For example, we could have the sequence (a_n) = (+\infty,-\infty,+\infty,-\infty,+\infty,\ldots), which diverges but has various subsequences converging to either +\infty or -\infty.

Functions

Just as we can deal with the limit of sequences in \overline{\mathbb{R}}, we can also deal with limits of functions. Most limits behave similarly or identically to those in \mathbb{R}, but at the extrema of the real line the properties of our full topological space come into play.

The definition of a functional limit is ultimately the same in the extended reals as in the ordinary reals: \lim_{x\to c} f(x) = L if for every neighborhood N of L, there exists a neighborhood M of x such that f(m) \in f(N) whenever m \in M except possibly when m = x. Recall that in the ordinary reals there is a notion of a ‘limit at infinity’, as well as a notion that a function can ‘tend to infinity’ as a limit. The typical definition for the first is that \lim_{x \to \pm\infty} f(x) = L if, for every neighborhood N of L, we can choose a real number M such that whenever n > M (or n < M for a limit at -\infty), f(n) \in N. For the second, we say that \lim_{x \to c} f(x) = \pm\infty if, for every real number M, we can choose a neighbohood M of x such that whenever n \neq x \in M, f(n) > M (or, again f(n) < M). These two definitions may seem somewhat arbitrary compared to the definition of limits defined on ordinary real numbers. However, we find that these two definitions are logically equivalent to the statements \lim_{x \to \pm\infty} f(x) = L and \lim_{x \to c} f(x) = \pm\infty in \overline{\mathbb{R}} using the ordinary definition of a functional limit; they are simply rephrased to avoid treating \pm\infty as points on the real number line. This is one simple example of how using \overline{\mathbb{R}} can simplify statements in analysis.

Continuity

Including \pm\infty as points in our topological space can also provide new concepts not usually considered in \mathbb{R}. A function defined on all of \overline{\mathbb{R}} must take on values at the endpoints of \pm\infty. And once we have defined the value of a function at these two points, we can consider questions like ‘given some function f, is f(x) continuous at \pm\infty?’ As in \mathbb{R}, we define continuity at a point c to simply mean that \lim_{x \to c} f(x) = f(c).

For a simple example, we can show that f(x) = x is continuous on all of \overline{\mathbb{R}}, including \pm\infty. We need to show that for every neighborhood N of f(x) there exists a neighborhood M of x such that f(M) \subseteq N. But because f(x) = x and f(M) = M we can just let M = N for every x and N. Because f(x) \in N and N \subseteq N, this is enough to show that f is continuous on \overline{\mathbb{R}}. Note that this proof is identical to one in \mathbb{R}; here, the fact that the function is also defined at \pm\infty does not effect the behavior of the function’s limits or continuity.

Dealing with a nonlinear function like g(x) = e^x (with the stipulation that g(-\infty) = 0 and g(+\infty) = +\infty) is slightly harder. Here we will only prove that the function is continuous at \pm\infty; continuity at points in \mathbb{R} can be shown using the usual methods of analysis on the reals. To show continuity at x = +\infty, we only need to show that for every neighborhood N of +\infty there is some neighborhood M of +\infty such that f(M) \subseteq N. Conveniently g is an injection, so there is a well-defined inverse g^{-1}: [0,+\infty] \to \overline{\mathbb{R}}. In fact, g^{-1} is simply the natural extension of \ln(x) to \overline{\mathbb{R}} where \ln(0) = -\infty and \ln(+\infty) = +\infty. So for any N we can let M = g^{-1}(N). The distinctive feature of a neighborhood of +\infty is that it contains an interval of the form (a,+\infty], and we find that g^{-1}((a,+\infty]) = (\ln(a),+\infty]. So M will still contain an open interval containing +\infty, and we have found our neighborhood. Therefore g is continuous at +\infty. A similar argument allows us to construct a neighborhood M of -\infty such that g(M) is a subset of a given neighorhood N of 0, showing that g is also continuous at -\infty.

The above f and g provide examples of functions which are everywhere continuous on \mathbb{R} which stay continuous on all of \overline{\mathbb{R}}. But there are other functions continuous on \mathbb{R} which cannot be extended in a way that makes them continuous at \pm\infty. For example, h(x) = \sin(x) will be discontinuous at both endpoints of the real line regardless of what values are assigned to \sin(\pm\infty). Any neighborhood of +\infty we choose will contain an interval of the form (a,+\infty] for some a, and h oscillates between \pm1 infinitely many times in this interval. So h has no limit at +\infty, and cannot be continuous there; an identical argument shows h must be discontinuous at -\infty as well.

Because \pm\infty are somewhat special points, questions about function behavior immediately come to mind. For example, is it possible to have f: \overline{\mathbb{R}} \to \overline{\mathbb{R}} be discontinuous on all of \mathbb{R}, but continuous at \pm\infty?

The answer to that particular question turns out to be yes, and it’s not much more difficult to construct a function continuous only at \pm\infty than it is to construct a function continuous at other isolated points. For example, consider the Dirichlet function, defined as f(x) = 1 if x \in \mathbb{Q} and f(x) = 0 otherwise. This is not only discontinuous on all of \mathbb{R}, but can also be shown to be discontinuous at \pm\infty regardless of what values are assigned to f(\pm\infty). Just like \sin(x), f oscillates between 0 and 1 infinitely often in any neighborhood of \pm\infty, and therefore cannot be continuous.

It is easily shown that g(x) = xf(x) is continuous at 0, but nowhere else in \mathbb{R}. We can flip this around to define h(x) = \frac{f(x)}{x} (specifying that h(+\infty) = h(-\infty) = 0) in order to produce a function continuous only at the endpoints of the real line. This works because the values of h(x) grow arbitrarily small for both rational and irrational values of x as x grows without bound.

We can also find functions continuous on all of \overline{\mathbb{R}} which would be discontinuous if defined on \mathbb{R}. The classic example is f(x) = \frac{1}{x^2}, which can be defined on all of \overline{\mathbb{R}} by setting f(+\infty) = f(-\infty) = 0 and f(0) = +\infty and which turns out to be continuous on the entire space. By contrast, this function is undefined at 0 when we are restricted to \mathbb{R} and, even if a value were assigned at 0, f could not possibly be continuous there. On the other hand g(x) = \frac{1}{x} will still be discontinuous at 0 even using the extended real number line, since \lim_{x \to 0^-} = -\infty while \lim_{x \to 0^+} = +\infty.

And More

The next obvious topic of investigation is derivatives. However, these cause far more problems than the rather elegant extensions of limits and continuity to \overline{\mathbb{R}}. Limits of sequences and continuous functions can be dealt with in any topological space; derivatives require some notion of distance. While it is possible to metrize the extended real line, there is no way to do so which agrees with the natural way of measuring distances on the ordinary real line, and so working with derivatives on \overline{\mathbb{R}} would be difficult or impossible.

There are also other ways to put endpoints on the real number line. For example, the projectively extended real line attaches a single endpoint \infty which acts as both an upper and lower bound for the reals. The open intervals are elegantly defined as the intervals (a,b), and we simply remove the requirement that a \le b. For example, (0,1) is the same interval as in \mathbb{R}, while (1,0) = (1,\infty) \cup \{\infty\} \cup (\infty,0); the interval ‘wraps around` infinity back to 0.

The affinely extended real line, projectively extended real line, and other topological spaces expand the notions of limits and continuity we are familiar with in \mathbb{R} while also shedding light on the behavior of the ordinary reals. While their structures are different, they are all useful ways of extending our familiar number system to analysis at the nonstandard ‘endpoints’ of the real number line.

Posted on by David Clayton
Categories: Fall 2016

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