nLab
topological field

Contents

Contents

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 SetSet is a field KK and such that the multiplicative inversion operation i:K{0}K{0}i: K \setminus \{0\} \to K \setminus \{0\} is continuous with respect to the subspace topology inherited from KK.

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 nn\mathbb{Z}, 0n0 \neq n \in \mathbb{Z}, to be a filter base for a filter of neighborhoods of 00. That this gives a ring topology follows from three easily verified facts: (1) for any ideal II there is an ideal JJ such that J+JIJ + J \subseteq I and JI-J \subseteq I (e.g., J=IJ = I), (2) if II is any ideal and qq \in \mathbb{Q} is any element, there is an ideal JJ such that qJIq J \subseteq I, and (3) if II is any ideal, there is an ideal JJ such that JJIJ J \subseteq I (again take J=IJ = 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 11 not containing 00, e.g., 1+21 + 2\mathbb{Z}, no neighborhood 1+m1 + m\mathbb{Z} of 11 will fit inside the set of reciprocals {11+2n:n}\{\frac1{1 + 2n}: n \in \mathbb{Z}\}.

Remark

If KK is a topological field, then either KK is a codiscrete space or is a Tychonoff space. The reason is that the closure of {0}\{0\} in a topological ring KK must be an ideal II, and since KK is a field, II is either all of KK (whence KK is codiscrete), or I={0}I = \{0\} in which case KK is a T 1T_1-space. In the latter case, since a topological ring is a uniform space, the T 1T_1-condition implies KK 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 TopTop by the fact that TopTop 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 TopTop, as well as in other categories of interest.

(Note that TopTop 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 TopSetTop \to Set is a topological functor. Curiously, Top opTop^{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 KK in terms of the following limit-colimit sketch:

  1. Introduce structure to make KK a commutative ring object: two binary operations a:K×KKa: K \times K \to K (addition) and m:K×KKm: K \times K \to K (multiplication), two constants 0:1K0: 1 \to K and e:1Ke: 1 \to K (additive and multiplicative identities), additive inversion :KK-: K \to K, all subject to the usual equations for commutative rings;

  2. Letting i:UK×Ki: U \to K \times K denote the equalizer of m:K×KKm: K \times K \to K and e!:K×K1Ke \circ !: K \times K \to 1 \to K, add the axiom that j=π 1i:UK×KKj = \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+UK(0, j): 1 + U \to K (provably a disjoint embedding) is epic.

Some commentary might be in order. Clearly UU plays the role of the group of units of KK, realized as a subobject by j:UKj: U \to K. Axiom 3. says that 00 and UU exhaust all of KK, but without going so far to say that (0,j):1+UK(0, j): 1 + U \to K is an isomorphism, an inappropriately strong condition in the case of TopTop (as it would force the point 0:1K0: 1 \to K to be open, making KK a discrete space).

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

Examples

etc.

Last revised on May 30, 2017 at 05:10:57. See the history of this page for a list of all contributions to it.