Example 2

Let $\Sigma$ be the species of topological structures (no. 4, Example 3), let $A$ be the (principal) base set, and let $V$ be the generic structure of $\Sigma$. Consider the relation

This relation has a graph $P \subset \mathfrak{P}(A) \times A$ with respect to the pair $(X, x)$; $P \mid A, V$ is a term called “the set of all pairs $(X, x)$ such that $x$ lies in the closure of $X$ with respect to the topology $V$”. We can then prove (cf. General Topology, Chapter I, § 1) that the following relations are theorems in $\mathscr{T}_{\Sigma}$:

Consider the species of structures $\Theta$, having a single (principal) base set $A$, whose generic structure consists of a single letter $W$, which has as typical characterization $W \in \mathfrak{P}(\mathfrak{P}(A) \times A)$ and as axiom

and

and

Consider also the relation

The set of all $U \in \mathfrak{P}(A)$ which satisfy this relation is a subset $Q \mid A, W$ of $\mathfrak{P}(A)$. We can then prove (General Topology, Chapter I, § 1, Exercise 10) that the following relations are theorems in $\mathscr{T}_{\Theta}$:

Thus the terms $P\{A,V\}$ and $Q\{A,W\}$ satisfy conditions (1) and (2) above, and it is easily seen that they also satisfy condition (3). The species of structures $\Sigma$ and $\Theta$ are therefore equivalent, and we therefore consider every structure of species $\Theta$ as a topology, namely that which corresponds to it under the procedure of deduction $Q\{A,W\}$. \*