I will try to provide some more intuition for the topology of uniform convergence.

As @hmakholm left over Monica wrote, this topology is given by the uniform metric.

This means that open sets can be "built" from open balls with all points have *distance* from the center of the circle lower than some radius. We can talk about distance because we can measure it by the metric.

To put it into context, uniform topology is a topology that is finer than the product topology but coarser than the box topology on that set. It can be useful for example to determine how functions on your space behave.

**Example**

Take the uniform topology on $\mathbb{R}^I$, where $I = \{f: I → \mathbb{R}\}$ is defined by

$\overline{d_∞}(f, g) =$ min $\{sup_{i∈I} |f(i) − g(i)|, 1\}$.

Then a sequence of functions $f_n$ converges to $f$ if and only if $\overline{d_∞}(f, g)$ converges to $0$.

I recommend book "Topology" by Munkers to see some more examples.