Automatic continuity (Part 1)

Given two mathematical objects $S_0$ and $S_1$ and a function $f: S_0 \rightarrow S_1$ between them, it is common to ask whether some or all of the mathematical structure of the objects is respected by the function $f$ . If the objects $S_0$ and $S_1$ are simply abstract sets then of course any function $f$ preserves the structure. When $S_0$ and $S_1$ are groups then the question as to whether $f$ preserves the underlying mathematical structure is resolved by considering whether $f(ab) = f(a)f(b)$ for all $a, b\in S_0$ ; that is, one checks whether $f$ is a homomorphism. More generally when the objects are rings, modules, universal algebras, or relational structures the relevant question is whether $f$ is a homomorphism (or possibly a strong homomorphism). When the objects are topological spaces one can ask whether $f$ is continuous. As objects become more laden with structure, questions regarding such a map $f$ can become more interesting or complicated.

Suppose the objects are topological groups- that is groups $G$ having a topology for which multiplication the multiplication function $G \times G \rightarrow G$ given by $(g, g') \mapsto gg'$ and the inverse function $G\rightarrow G$ given by $g \mapsto g^{-1}$ are continuous. One can ask whether preservation of the algebraic structure implies preservation of the topological structure: Is a group homomorphism continuous? It would be unimaginable to expect each group homomorphism to be continuous, irrespective of the domain and the codomain. As a cheap example one can take $S_0$ and $S_1$ to be the same group but with $S_1$ having a finer topology (more open sets) than $S_0$ . The identity map $\iota: S_0 \rightarrow S_1$ is an isomorphism but is obviously not continuous. More concretely we can let $S_0$ and $S_1$ be the additive group on the real numbers and endow $S_0$ with the standard topology and $S_1$ with the discrete topology.

For a more interesting example one can take the domain and codomain to again be the additive group on the reals and endow both with the standard topology. By considering $\mathbb{R}$ as a vector space over the field $\mathbb{Q}$ and selecting a basis including $1\in \mathbb{R}$ , we can express the group isomorphism $\mathbb{R} \simeq \bigoplus_{2^{\aleph_0}}\mathbb{Q}$ . We may project to the coordinate associated with the basis element $1$ , giving us a homomorphic retraction $f: \mathbb{R}\rightarrow \mathbb{Q} \leq \mathbb{R}$ . This homomorphism is obviously not continuous since the domain is connected and the image is totally disconnected. More generally one can prove (see Srivastava’s A Course on Borel Sets, Exercise 3.5.11) that any continuous (or Borel measurable, or even Baire measurable) homomorphism from $\mathbb{R}$ to itself is given by multiplication by a constant. Thus one obtains many examples of discontinuous automorphisms of $\mathbb{R}$ by permuting the coordinates of $\bigoplus_{2^{\aleph_0}}\mathbb{Q}$ .

For another example, suppose that the mathematical structures are path connected topological spaces $X$ and $Y$ and we have an abstract group homomorphism $\phi: \pi_1(X, x) \rightarrow \pi_1(Y, y)$ . It is natural to ask whether there exists a continuous function $f: (X, x) \rightarrow (Y, y)$ such that $f_*: \pi_1(X, x) \rightarrow \pi_1(Y, y)$ equals $\phi$ . This gives a slightly different sense of continuity of a homomorphism. In the next post we will give some concrete examples of cases where homomorphisms are “automatically continuous.”

Share this:

Like this:

Related

Leave a Reply Cancel reply