Stone-Weierstrass Theorem/Also presented as

Stone-Weierstrass Theorem: Also presented as

Some sources specifically define $T = \struct {X, \tau}$ to be a Hausdorff space.

However, we note that by hypothesis $\AA$ is such that it separates points.

Hence from Topological Space Separated by Mappings is Hausdorff it follows that $T$ is a fortiori a Hausdorff space.