Topology of uniform convergence?
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 $X\to\mathbb R$.