# Continuous functions from compact Hausdorff spaces to the interval

Let $$X$$ be a compact Hausdorff space and let $$C(X,I)$$ be the set of all continuous functions from $$X$$ into the closed interval $$[0,1]$$. If we equip $$C(X,I)$$ with the the topology of uniform convergence is $$C(X,I)$$ compact? I'm inclined to think not, but I can't think of a simple counterexample.

$$C(X,I)$$ is the closed ball of radius $$1/2$$ centred at the constant function $$1/2$$ in the Banach space $$C(X,\mathbb R)$$. A closed ball of nonzero radius in a Banach space is compact if and only if the Banach space is finite-dimensional. $$C(X,\mathbb R)$$ is infinite-dimensional. Therefore $$C(X,I)$$ is not compact.