Remark

We have just met an example (Example 2) in which two comparable structures of the same species $\Sigma$ are necessarily identical. There are many other such examples : linear order structures, \* compact topologies, Fréchet space structures (the morphisms being continuous linear mappings), topologies defined by an absolute value (or a valuation) on a division ring, etc. \*

For such a species of structures $\Sigma$, a morphism $f$ of E into E' which is bijective is an isomorphism; for if we transport to E' the structure $\mathscr{S}$ on E by means of $f$, we obtain a structure of species $\Sigma$ which is finer than the structure $\mathscr{S}'$ on E' and therefore coincides with $\mathscr{S}'$.