Definition 6.1.4 (Pointwise Topology). Let $T$ be a set and $X$ be a topological space, then the following topologies on $X^{T}$ coincide:
The product topology on $X^{T}$.
The $\mathfrak{S}$-open topology, where $\mathfrak{S}= \bracs{\bracs{x}| x \in X}$.
(If $X$ is a uniform space) The $\mathfrak{F}$-uniform topology, where $\fF = \bracs{F| F \subset X \text{ finite}}$.
This topology is the topology of pointwise convergence on $X^{T}$.
Proof. (2) $=$ (3): Let $F \subset X$ finite and $U$ be an entourage, $f \in X^{T}$, then
\[E(F, U)(f) = \bigcap_{x \in F}\pi_{x}^{-1}(U(f(x)))\]
which is open in the product topology. The converse is given by Definition 6.1.2.$\square$