Remark

For given species of structures $\Sigma$ we may have occasion to define various terms $\sigma \} x,\ y,\ s,\ t \{$ which satisfy the conditions $(\mathrm{MO_I})$, $(\mathrm{MO_{II}})$, and $(\mathrm{MO_{III}})$. For example, if $\Sigma$ is the species of topological structures, with the notation of Example 3 above, a mapping $f$ of A into A$'$ is said to be open if the relation $\mathrm{X} \in \mathrm{V}$ implies $f(\mathrm{X}) \in \mathrm{V}'$ (in other words, if the image under $f$ of every open set is an open set). It is easily checked that the open mappings also satisfy conditions $(\mathrm{MO_I})$, $(\mathrm{MO_{II}})$, and $(\mathrm{MO_{III}})$ for the species $\Sigma$. \* Moreover, it can be shown that a continuous mapping is not necessarily open, and that an open mapping is not necessarily continuous. \* A given species of structures therefore does not imply a well-defined notion of morphisms.

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Where order structures, algebraic structures, and topological structures are concerned, it is always to be understood that the morphisms are those which have been defined in the Examples above, unless the contrary is expressly stated.

The condition $(\mathrm{MO_{III}})$ and the characterization of bijections (Chapter II, § 3, no. 8, Corollary to Proposition 8) imply the following criterion :

CST8. Let E, E$'$ be two sets, each endowed with a structure of species $\Sigma$. Let $f$ be a $\sigma$-morphism of E into E$'$ and let $g$ be a $\sigma$-morhpism of E$'$ into E. If $g \circ f$ is the identity mapping of E onto itself, and if $f \circ g$ is the identity mapping of E$'$ onto itself, then $f$ is an isomorphism of E onto E$'$, and $g$ is the inverse isomorphism.

It should be noted that a bijection of E onto E$'$ may be a $\sigma$-morphism without the inverse bijection necessarily being a $\sigma$-morphism. \* For example, a bijective mapping of a topological space A onto a topological space A$'$ can be continuous without the inverse bijection being continuous (General Topology, Chapter I, § 2, no. 1, Remark 1). \*

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