Proposition 2

If to each element $x$ of a set $X$ there corresponds a set $\mathfrak{B}(x)$ of subsets of $X$ such that the properties (VI), (V_{II}), (V_{III}) and (V_{IV}) are satisfied, then there is a unique topological structure on $X$ such that, for each $x \in X$, $\mathfrak{B}(x)$ is the set of neighbourhoods of $x$ in this topology.*

By Proposition 1, if there is a topology on $X$ satisfying these conditions, the set of open sets for this topology is necessarily the set $\mathscr{D}$ of subsets $A$ of $X$ such that for each $x \in A$ we have $A \in \mathfrak{B}(x)$; hence the uniqueness of this topology if it exists.

The set $\mathscr{D}$ certainly satisfies axioms (O_I) and (O_{II}): for (O_I), this follows immediately from (V_I), and for (O_{II}), from (V_{II}). It

!Diagram showing sets V, U, W, and points x, y, z

Figure 1.

remains to show that, in the topology defined by $\mathscr{D}$, $\mathfrak{B}(x)$ is the set of neighbourhoods of $x$ for each $x \in X$. It follows from (V_I) that every neighbourhood of $x$ belongs to $\mathfrak{B}(x)$. Conversely, let $V$ be a set belonging to $\mathfrak{B}(x)$, and let $U$ be the set of points $y \in X$ such that $V \in \mathfrak{B}(y)$; if we can show that $x \in U, U \subset V$ and $U \in \mathscr{D}$, then the proof will be complete. We have $x \in U$ since $V \in \mathfrak{B}(x)$; also $U \subset V$, for every point $y \in U$ belongs to $V$ by reason of (V_III) and the hypothesis $V \in \mathfrak{B}(y)$. It remains to show that $U \in \mathscr{D}$, i.e. that $U \in \mathfrak{B}(y)$ for each $y \in U$; now (Fig. 1) if $y \in U$ then by (V_IV) there is a set $W$ such that for each $z \in W$ we have $V \in \mathfrak{B}(z)$; since $V \in \mathfrak{B}(z)$ means that $z \in U$, it follows that $W \subset U$, and therefore, by (V_I), that $U \in \mathfrak{B}(y)$.

Q.E.D.

Proposition 2 shows that a topology on $X$ can be defined by means of the sets $\mathfrak{B}(x)$ of neighbourhoods of points of $X$, subject only to the axioms (V_I), (V_II), (V_III) and (V_IV).