Exercise 8, § 2

Let $\Sigma$ be the species of structures which has a single (principal) base set, whose generic structure (V, $a$, $b$) consists of three letters, with the typical characterization

and whose axiom is the relation

where $\mathrm{R}\{\mathrm{V}\}$ is the axiom of the species of topological structures (§ 1, no. 4, Example 3). If A, A$'$ are two sets endowed respectively with structures (V, $a$, $b$), (V$'$, $a'$, $b'$) of species $\Sigma$, the $\sigma$-morphisms of A into A$'$ are defined to be the continuous mappings $f: \mathrm{A} \to \mathrm{A}'$ (with respect to the topologies V, V$'$) such that $f(a) = a'$ and $f(b) = b'$. Show that by replacing $\Sigma$ by an equivalent species of structures, we can obtain this notion of $\sigma$-morphisms by the procedure of Exercise 2. Show that if A, B are two sets endowed with structures S, S$'$ of species $\Sigma$, there exists a product structure on $\mathrm{A} \times \mathrm{B}$, and that the direct image of this structure under $\mathrm{pr}_1$ (resp. $\mathrm{pr}_2$) is S (resp. S$'$). \* But give an example where there is no section of $\mathrm{pr}_1$ which is a $\sigma$-morphism of A into $\mathrm{A} \times \mathrm{B}$ (take A to be connected and B discrete). \*