Proposition 7.2.3.label Let $T$ be a set, $\sigma \subset 2^{T}$ be an ideal, and $(X, \fU)$ be a uniform space whose uniformity is induced by the pseudometrics $\seqi{d}$. For each $i \in I$ and $S \in \sigma$, let
then
- (1)
$\bracs{d_{i, S}| i \in I, S \in \sigma}$ is a family of pseudometrics induces the $\sigma$-uniformity on $X^{T}$.
- (2)
The family
\[\bracs{\bigcap_{j \in J}E(d_{j, S}, r)|J \subset I \text{ finite}, r > 0, S \in \sigma}\]is a fundamental system of entourages for the $\sigma$-uniformity on $X^{T}$.
Proof. (1): Let $S \in \sigma$ and $U \in \fU$, then there exists $r > 0$ and $J \subset I$ finite such that $\bigcap_{j \in J}E(d_{j}, r) \subset U$, so
and the uniformity induced by $\bracs{d_{i, S}| i \in I, S \in \sigma}$ contains the $\sigma$-uniformity.
On the other hand, for any $i \in I$ and $r > 0$, $E(d_{j}, r/2) \in \fU$ by Definition 6.3.3. Therefore $E(S, E(d_{j}, r/2)) \subset E(d_{j, S}, r)$, so the $\sigma$-uniformity contains the induced uniformity.
(2): By Definition 7.2.2,
is a fundamental system of entourages for the $\sigma$-uniformity. Following the same steps in (1),
is a fundamental system of entourages for the $\sigma$-uniformity.$\square$
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