The Unapologetic Mathematician

The Topological Field of Real Numbers

We’ve defined the topological space we call the real number line $\mathbb{R}$ as the completion of the rational numbers $\mathbb{Q}$ as a uniform space. But we want to be able to do things like arithmetic on it. That is, we want to put the structure of a field on this set. And because we’ve also got the structure of a topological space, we want the field operations to be continuous maps. Then we’ll have a topological field, or a “field object” (analogous to a group object) in the category $\mathbf{Top}$ of topological spaces.

Not only do we want the field operations to be continuous, we want them to agree with those on the rational numbers. And since $\mathbb{Q}$ is dense in $\mathbb{R}$ (and similarly $\mathbb{Q}\times\mathbb{Q}$ is dense in $\mathbb{R}\times\mathbb{R}$), we will get unique continuous maps to extend our field operations. In fact the uniqueness is the easy part, due to the following general property of dense subsets.

Consider a topological space $X$ with a dense subset $D\subseteq X$. Then every point $x\in X$ has a sequence $x_n\in D$ with $\lim x_n=x$. Now if $f:X\rightarrow Y$ and $g:X\rightarrow Y$ are two continuous functions which agree for every point in $D$, then they agree for all points in $X$. Indeed, picking a sequence in $D$ converging to $x$ we have
$f(x)=f(\lim x_n)=\lim f(x_n)=\lim g(x_n)=g(\lim x_n)=g(x)$.

So if we can show the existence of a continuous extension of, say, addition of rational numbers to all real numbers, then the extension is unique. In fact, the continuity will be enough to tell us what the extension should look like. Let’s take real numbers $x$ and $y$, and sequences of rational numbers $x_n$ and $y_n$ converging to $x$ and $y$, respectively. We should have
$s(x,y)=s(\lim x_n,\lim y_n)=s(\lim(x_n,y_n))=\lim x_n+y_n$
but how do we know that the limit on the right exists? Well if we can show that the sequence $x_n+y_n$ is a Cauchy sequence of rational numbers, then it must converge because $\mathbb{R}$ is complete.

Given a rational number $r$ we must show that there exists a natural number $N$ so that $\left|(x_m+y_m)-(x_n+y_n)\right| for all $m,n\geq N$. But we know that there’s a number $N_x$ so that $\left|x_m-x_n\right|<\frac{r}{2}$ for $m,n\geq N_x$, and a number $N_y$ so that $\left|y_m-y_n\right|<\frac{r}{2}$ for $m,n\geq N_y$. Then we can choose $N$ to be the larger of $N_x$ and $N_y$ and find
$\left|(x_m-x_n)+(y_m-y_n)\right|\leq\left|x_m-x_n\right|+\left|y_m-y_n\right|<\frac{r}{2}+\frac{r}{2}=r$
So the sequence of sums is Cauchy, and thus converges.

What if we chose different sequences $x'_n$ and $y'_n$ converging to $x$ and $y$? Then we get another Cauchy sequence $x'_n+y'_n$ of rational numbers. To show that addition of real numbers is well-defined, we need to show that it’s equivalent to the sequence $x_n+y_n$. So given a rational number $r$ does there exist an $N$ so that $\left|(x_n+y_n)-(x'_n+y'_n)\right| for all $n\geq N$? This is almost exactly the same as the above argument that each sequence is Cauchy! As such, I’ll leave it to you.

So we’ve got a continuous function taking two real numbers and giving back another one, and which agrees with addition of rational numbers. Does it define an Abelian group? The uniqueness property for functions defined on dense subspaces will come to our rescue! We can write down two functions from $\mathbb{R}\times\mathbb{R}\times\mathbb{R}$ to $\mathbb{R}$ defined by $s(s(x,y),z)$ and $s(x,s(y,z))$. Since $s$ agrees with addition on rational numbers, and since triples of rational numbers are dense in the set of triples of real numbers, these two functions agree on a dense subset of their domains, and so must be equal. If we take the ${0}$ from $\mathbb{Q}$ as the additive identity we can also verify that it acts as an identity real number addition. We can also find the negative of a real number $x$ by negating each term of a Cauchy sequence converging to $x$, and verify that this behaves as an additive inverse, and we can show this addition to be commutative, all using the same techniques as above. From here we’ll just write $x+y$ for the sum of real numbers $x$ and $y$.

What about the multiplication? Again, we’ll want to choose rational sequences $x_n$ and $y_n$ converging to $x$ and $y$, and define our function by
$m(x,y)=m(\lim x_n,\lim y_n)=m(\lim(x_n,y_n))=\lim x_ny_n$
so it will be continuous and agree with rational number multiplication. Now we must show that for every rational number $r$ there is an $N$ so that $\left|x_my_m-x_ny_n\right| for all $m,n\geq N$. This will be a bit clearer if we start by noting that for each rational $r_x$ there is an $N_x$ so that $\left|x_m-x_n\right| for all $m,n\geq N_x$. In particular, for sufficiently large $n$ we have $\left|x_n\right|<\left|x_N\right|+r_x$, so the sequence $x_n$ is bounded above by some $b_x$. Similarly, given $r_y$ we can pick $N_y$ so that $\left|y_m-y_n\right| for $m,n\geq N_y$ and get an upper bound $b_y\geq y_n$ for all $n$. Then choosing $N$ to be the larger of $N_x$ and $N_y$ we will have
$\left|x_my_m-x_ny_n\right|=\left|(x_m-x_n)y_m+x_n(y_m-y_n)\right|\leq r_xb_y+b_xr_y$
for $m,n\geq N$. Now given a rational $r$ we can (with a little work) find $r_x$ and $r_y$ so that the expression on the right will be less than $r$, and so the sequence is Cauchy, as desired.

Then, as for addition, it turns out that a similar proof will show that this definition doesn’t depend on the choice of sequences converging to $x$ and $y$, so we get a multiplication. Again, we can use the density of the rational numbers to show that it’s associative and commutative, that $1\in\mathbb{Q}$ serves as its unit, and that multiplication distributes over addition. We’ll just write $xy$ for the product of real numbers $x$ and $y$ from here on.

To show that $\mathbb{R}$ is a field we need a multiplicative inverse for each nonzero real number. That is, for each Cauchy sequence of rational numbers $x_n$ that doesn’t converge to ${0}$, we would like to consider the sequence $\frac{1}{x_n}$, but some of the $x_n$ might equal zero and thus throw us off. However, there can only be a finite number of zeroes in the sequence or else ${0}$ would be an accumulation point of the sequence and it would either converge to ${0}$ or fail to be Cauchy. So we can just change each of those to some nonzero rational number without breaking the Cauchy property or changing the real number it converges to. Then another argument similar to that for multiplication shows that this defines a function from the nonzero reals to themselves which acts as a multiplicative inverse.

December 3, 2007 -

1. […] Order on the Real Numbers We’ve defined the real numbers as a topological field by completing the rational numbers as a uniform space, and then extending […]

Pingback by The Order on the Real Numbers « The Unapologetic Mathematician | December 4, 2007 | Reply

2. […] Spaces and continuity of real-valued functions Now that we’ve got the real numbers which correspond to our usual notion of magnitudes like distances, let’s refine our concept […]

Pingback by Metric Spaces and continuity of real-valued functions « The Unapologetic Mathematician | December 10, 2007 | Reply

3. […] in Series I As we’ve said before, the real numbers are a topological field. The fact that it’s a field means, among other things, that it comes equipped with an […]

Pingback by Associativity in Series I « The Unapologetic Mathematician | May 6, 2008 | Reply

4. Given that the theory of fields is not purely algebraic, what exactly constitutes a field object? That is, what would correspond to the condition that reciprocation’s domain should be precisely the non-zero elements? Do we need to work in a category with a notion of subobject complement or something like that?

Comment by Sridhar Ramesh | July 7, 2008 | Reply

5. I’m not sure offhand what you mean about the theory of fields not being algebraic. The theory of topological fields isn’t…

Comment by John Armstrong | July 7, 2008 | Reply

6. Sure, I understand what a topological field is; I’m just curious what a field object would amount to in general, in other categories.

Sorry, what I meant by “the theory of fields is not purely algebraic” was that, as a result of the stipulation “Reciprocation is defined for all and only the non-zero elements”, it wasn’t specifiable purely in terms of universal equations over some language of (total) operators, in the manner of universal algebra, the way groups, rings, vector spaces, etc., are; as a result, my simple understanding of “X object” in such cases [as a monoidal functor from a particular monoidal category whose structure is determined by those identities] couldn’t work as is.

Comment by Sridhar Ramesh | July 7, 2008 | Reply

7. Just to make my question clear, it is this: what exactly is the definition of a field object (where this definition is presumably interpretable in contexts beyond simply Set and Top)?

I would take it to be something like a pair of objects F and F* in a monoidal category, corresponding to the whole field and its multiplicative group, along with morphisms corresponding to the various field operators, satisfying the field identities _and_ satisfying some property along the lines of “F* should act as though it is all and only the nonzero elements of F”. The category theoretic-version of the property in quotes is what I can’t figure out.

Comment by Sridhar Ramesh | July 7, 2008 | Reply

8. Whoops, I shouldn’t have listed “vector spaces” above as an example of case where I already understand what “X object” would mean. Replace it with “modules”.

Comment by Sridhar Ramesh | July 7, 2008 | Reply

9. Ramesh, as you point out, the notion of field is not algebraic (i.e., is not given by a Lawvere algebraic theory, or for that matter by a suitable monad). To define the notion of a field in a category C, you therefore need to assume more of C than that it just has cartesian products.

A more or less satisfactory solution is to assume that C can support a certain amount of first-order logic, e.g., C is a pretopos or logos (see here for a quick sketch of the relevant notions), or more strongly, a topos. A lot of it boils down to certain assumptions on the structure of subobject lattices (for example, in a logos the subobject lattices form Heyting algebras, so that one can interpret operations like implication and negation), and on operations between them (pulling back or “substitution” operations, and pushing forward or “quantification” operations adjoint to pulling back, e.g., direct image operations as left adjoint to pulling back).

Probably a good place to begin learning about internalizing first-order logic in a category is by reading the book on topos theory by Mac Lane and Moerdijk.

Even so, there are subtleties because in the typical applications, pretoposes or logoses or toposes tend to be “intuitionistic”, i.e., the subobject lattices are not generally Boolean algebras, but Heyting algebras. For example, we could define a field by interpreting

forall x (not x = 0) => exists_y xy = 1

or by interpreting say

forall x (x = 0) or (exists_y xy = 1)

but in intuitionistic logic, the latter formulation will be strictly stronger than the former. Then one has to decide which is more “appropriate” for the application at hand (which I won’t get into here).

Comment by Todd Trimble | July 8, 2008 | Reply

10. Ah, alright. I have some familiarity with topos theory (and no qualms with intuitionistic logic, which has pretty much became my default manner of thinking), but I was wondering if something less heavy-duty would suffice.

It looks from your link as though a prelogos should be more than able to handle something like “for all x. (x = 0) or (the inverse of x is defined)”, since it seems to have the right structure to deal with disjunctions. I guess a logos would be able to go further and also handle “forall x. (not x = 0) (the inverse of x is defined)” [my preferred formulation], since Lawvere-style generalized universal quantification and the empty disjunction should let us express “the object of non-zero elements” fairly directly.

It’s a shame that this all seems still quite a bit heavier than “monoidal categories” or “categories with finite products”; I suppose one could pare down the structure of a logos into only what is needed in this case, but perhaps it wouldn’t be very natural to do so. Ah well.

Comment by Sridhar Ramesh | July 8, 2008 | Reply

11. I had written a bi-implication operator above whose angle brackets seem to have doomed it to be parsed as some kind of HTML tag instead. The second quote in the second paragraph above should read “forall x. (not x = 0) iff (the inverse of x is defined)”.

Comment by Sridhar Ramesh | July 8, 2008 | Reply

12. Looking back on this discussion, it occurs to me that Top is none of the kinds of logic-interpreting categories mentioned above (as it is not regular). But its subobject lattices will have pseudocomplements, I suppose, which should be enough for defining field objects.

Comment by Sridhar Ramesh | August 1, 2008 | Reply

13. […] real numbers have a topology. In fact, that’s really their main characteristic. The rational numbers have a topology too, […]

Pingback by Some Topological Fields « The Unapologetic Mathematician | August 26, 2008 | Reply