Exercise 7, § 2

Let $\Sigma$ be the species of structures which has a principal base set A, and the set $\mathbf{R}$ as auxiliary base set, whose generic structure consists of two letters $(\mathrm{V}, \varphi)$ with the typical characterization

and whose axioms are (i) the axiom $\mathrm{R}\{\mathrm{V}\}$ of the species of topological structures (§ 1, no. 4, Example 3), (ii) the relation "there exists $a > 0$ such that $\varphi$ is the graph of a continuous injective mapping (with respect to the topology V) of the interval $[0, a]$ into A".

If A, A$'$ are two sets endowed respectively with structures $(\mathrm{V}, \varphi)$, $(\mathrm{V}', \varphi')$ of the species $\Sigma$, the $\sigma$-morphisms of A into A$'$ are defined to be the mappings $f$ of A into A$'$ which are continuous (with respect to the topologies V, V$'$) and whose graph F is such that $\mathrm{F} \circ \varphi \subset \varphi'$. Show that this notion of $\sigma$-morphisms can be defined by the procedure of Exercise 2, and that there exists a product structure on the product of two sets A$_1$, A$_2$ endowed with arbitrary structures of species $\Sigma$. But give an example in which the direct image, under the first projection $\mathrm{pr}_1$, of the product structure on $\mathrm{A}_1 \times \mathrm{A}_2$ is not the original structure given on A$_1$ (take A$_1$ to be a space homeomorphic to $\mathbf{R}$). \*