Topology of uniform convergence?

Question

Stone starts with an arbitrary compact Hausdorff space X and considers the algebra C(X,R) of real-valued continuous functions on X, with the topology of uniform convergence

I am having a hard time understanding what the topology of uniform convergence means. I of course know what uniform convergence means - a sequence of functions satisfying a particular set of convergence properties - but the "topology of uniform convergence"? Not sure.

Answer 1

I would assume it means to view $C(X,\mathbb R)$ as a metric space with the uniform metric $$d(f,g)=\sup_{x\in X}\;|f(x)-g(x)|$$ and derive a topology from that metric. Then convergence of a sequence under this toplogy is the same as uniform convergence of functions $X\to\mathbb R$.

Answer 2

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.

Answer 3

The “topology of uniform convergence” is the unique topology with the property that a set $A$ is closed if and only if for any uniformly convergent sequence $x_n\in A$ the limit point belongs to $A$.

It is a trivial observation that this is exactly the topology induced by sup-distance.

Indeed, $A$ has the property that it contains all limit points of uniformly convergent sequences $x_n\in A$ if and only if $A$ is closed with respect to the sup-distance.