nLab topological field

Idea

A topological field is a field equipped with a topology such that all of the field operations are continuous functions.

Definition

A topological field is a topological ring whose underlying ring in $Set$ is a field $K$ and such that the multiplicative inversion operation $i: K \setminus \{0\} \to K \setminus \{0\}$ is continuous with respect to the subspace topology inherited from $K$.

Remark

Note it is not automatic that inversion is continuous for a field equipped with a ring topology (when it is, we say the ring topology is a field topology). The following gives an example of a ring topology that is not a field topology: on the rational numbers $\mathbb{Q}$, take the set of ideals $n\mathbb{Z}$, $0 \neq n \in \mathbb{Z}$, to be a filter base for a filter of neighborhoods of $0$. That this gives a ring topology follows from three easily verified facts: (1) for any ideal $I$ there is an ideal $J$ such that $J + J \subseteq I$ and $-J \subseteq I$ (e.g., $J = I$), (2) if $I$ is any ideal and $q \in \mathbb{Q}$ is any element, there is an ideal $J$ such that $q J \subseteq I$, and (3) if $I$ is any ideal, there is an ideal $J$ such that $J J \subseteq I$ (again take $J = I$). It’s easy to see that this ring topology is Hausdorff.

But inversion on the nonzero elements is not continuous. Indeed, for a neighborhood of $1$ not containing $0$, e.g., $1 + 2\mathbb{Z}$, no neighborhood $1 + m\mathbb{Z}$ of $1$ will fit inside the set of reciprocals $\{\frac1{1 + 2n}: n \in \mathbb{Z}\}$.

Remark

If $K$ is a topological field, then either $K$ is a codiscrete space or is a Tychonoff space. The reason is that the closure of $\{0\}$ in a topological ring $K$ must be an ideal $I$, and since $K$ is a field, $I$ is either all of $K$ (whence $K$ is codiscrete), or $I = \{0\}$ in which case $K$ is a $T_1$-space. In the latter case, since a topological ring is a uniform space, the $T_1$-condition implies $K$ is a Tychonoff space.

Internalization

As discussed at field, the notion of field is not algebraic in the sense of algebraic theory, and there are various inequivalent ways of attempting to internalize the notion inside structured categories. This issue is compounded in $Top$ by the fact that $Top$ has few of the exactness properties one needs to enact the more “traditional” fragments of first-order logic (such as being a pretopos or Heyting category or exact category or regular category). So it is a matter of interest to give a categorical definition of field that internalizes correctly in $Top$, as well as in other categories of interest.

(Note that $Top$ does enjoy some elementary exactness properties: it is a lextensive category with finite colimits. It also satisfies a strong non-elementary condition: it is $\infty$-extensive and the underlying-set functor $Top \to Set$ is a topological functor. Curiously, $Top^{op}$ is a regular category.)

“Classical” internalization

One straightforward approach, at least if we are thinking along lines of classical logic, is to define a field $K$ in terms of the following limit-colimit sketch:

1. Introduce structure to make $K$ a commutative ring object: two binary operations $a: K \times K \to K$ (addition) and $m: K \times K \to K$ (multiplication), two constants $0: 1 \to K$ and $e: 1 \to K$ (additive and multiplicative identities), additive inversion $-: K \to K$, all subject to the usual equations for commutative rings;

2. Letting $i: U \to K \times K$ denote the equalizer of $m: K \times K \to K$ and $e \circ !: K \times K \to 1 \to K$, add the axiom that $j = \pi_1 \circ i: U \to K \times K \to K$ (provably monic in finite limit logic) is a regular monomorphism: the equalizer of its cokernel pair;

3. Add the axiom that $(0, j): 1 + U \to K$ is a monic epic.

Some commentary might be in order. Clearly $U$ plays the role of the group of units of $K$, realized as a subobject by $j: U \to K$. Axiom 3. says that $0$ and $U$ exhaust all of $K$, but without going so far to say that $(0, j): 1 + U \to K$ is an isomorphism, an inappropriately strong condition in the case of $Top$ (as it would force the point $0: 1 \to K$ to be open, making $K$ a discrete space).

Axiom 2. is more subtle: a mono $k: A \to B$ in $Top$ is regular iff $A$ has the subspace topology inherited from $B$ via $k$. So Axiom 2. interpreted in $Top$ says that the subspace topology on $U$ coming from its inclusion into $K$ coincides with the topology it has by definition, viz. the subspace topology coming from its embedding $i$ in $K \times K$. Notice that inversion on $U$ is continuous if we use the definitional topology, since inversion is effected by permuting the two factors of $K \times K$. Thus Axiom 2. is a sneaky way of forcing inversion on $U$ with the subspace topology from $K$ to be continuous (and in fact it is equivalent to continuity of inversion).

Examples

• The real numbers with their metric topology as a Euclidean space;

• The complex numbers, similarly,

etc.