# Topology of uniform convergence?

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.

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 $X\to\mathbb R$.
• I get it. Whoever wrote the Wikipedia article misuses "topology of uniform convergence" to mean "topology induced by $\|\cdot\|_\infty$". How terrible. This individual should be banned from writing about mathematics. – Rudy the Reindeer Jul 14 '14 at 11:09
• The corresponding wiki seems have been changed, though I was lost reading it. Are there other concerns than the continuous functions may not be bounded? Do they coincide when we consider $C_c (X)$ instead? – Syl.Qiu Jan 6 '17 at 7:29