Given two mathematical objects and and a function between them, it is common to ask whether some or all of the mathematical structure of the objects is respected by the function . If the objects and are simply abstract sets then of course any function preserves the structure. When and are groups then the question as to whether preserves the underlying mathematical structure is resolved by considering whether for all ; that is, one checks whether is a homomorphism. More generally when the objects are rings, modules, universal algebras, or relational structures the relevant question is whether is a homomorphism (or possibly a strong homomorphism). When the objects are topological spaces one can ask whether is continuous. As objects become more laden with structure, questions regarding such a map can become more interesting or complicated.
Suppose the objects are topological groups- that is groups having a topology for which multiplication the multiplication function given by and the inverse function given by 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 and to be the same group but with having a finer topology (more open sets) than . The identity map is an isomorphism but is obviously not continuous. More concretely we can let and be the additive group on the real numbers and endow with the standard topology and 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 as a vector space over the field and selecting a basis including , we can express the group isomorphism . We may project to the coordinate associated with the basis element , giving us a homomorphic retraction . 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 to itself is given by multiplication by a constant. Thus one obtains many examples of discontinuous automorphisms of by permuting the coordinates of .
For another example, suppose that the mathematical structures are path connected topological spaces and and we have an abstract group homomorphism . It is natural to ask whether there exists a continuous function such that equals . 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.”