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.”

 

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