Example 3

Take $\Sigma$ to be the species of topological strcutures (§1, no. 4, Example 3). Let A, A$'$ be two sets endowed with topologies V, V$'$, respectively. Consider the mappings $f$ of A into A$'$ such that the relation $\mathrm{X}' \in \mathrm{V}'$ implies $\overset{-1}{f}(\mathrm{X}') \in \mathrm{V}$ (in other words, such that the inverse image of every open set in the topology V$'$ is an open set in the topology V). These mappings, which satisfy $(\mathrm{MO_I})$, $(\mathrm{MO_{II}})$, and $(\mathrm{MO_{III}})$, are the continuous mappings of A into A$'$ (with respect to the topologies V and V$'$) (cf. General Topology, Chapter I, § 2).