Topology of pointwise convergence

Remarks

Viewing $Y^X$ as the product $\prod_{x \in X} Y$, this topology is exactly the product topology. A sequence $(f_n)$ converges to $f$ in it if and only if $f_n(x) \to f(x)$ in $Y$ for every $x \in X$, which is pointwise convergence. The same holds for nets.

Example: on $\R^{[0, 1]}$, let $f_n(x) = x^n$, and let $f(x) = 0$ for $x < 1$ and $f(1) = 1$. For each fixed $x$, the numbers $x^n$ tend to $f(x)$, so $f_n \to f$ in this topology.

Non-example: the discrete topology on $\R^{[0, 1]}$ is not the topology of pointwise convergence. In the discrete topology, the sequence $f_n$ above does not converge, because $\set{f}$ is open and contains no $f_n$.

For an uncountable set $X$ and $Y = \R$, the space $\R^X$ with this topology is not first-countable, so it is not metrizable.