Corollary 1

If $A$ is any subset of $X$ and $V$ is any symmetric entourage of $X$, then $V(A)$ is a neighbourhood of $A$ in $X$, and

where $\mathfrak{U}$ denotes the set of all entourages in $X$.

If $M = A \times A$, then $VMV = V(A) \times V(A)$ for any $V \in \mathscr{S}$; for the relation "there exists $p \in A$ such that $(x, p) \in V$" is by definition equivalent to $x \in V(A)$. The corollary now follows from Chapter I, § 4, no. 2, Proposition 5 and no. 3, Proposition 7.

$V(A)$ is said to be the V-neighbourhood of $A$.

If $V$ is an entourage which is open in $X \times X$, then $V(x)$ is open in $X$ for each $x \in X$ (Chapter I, § 4, no. 2, Corollary to Proposition 4) and therefore $V(A)$, being the union of the $V(x)$ as $x$ runs through $A$, is open in $X$. On the other hand, if $V$ is a closed entourage in $X \times X$, $V(A)$ need not be closed in $X$ for every subset $A$ of $X$ (Exercise 3).

It should also be remarked that as $V$ runs through the set of entourages of $X$, the sets $V(A)$ do not necessarily form a fundamental system of neighbourhoods of $A$ in $X$ (Exercise 2).