Exercise 26, § 9

If $X$ is a discrete space, show that the spaces $X''$ and $\tilde{X}$ defined in Exercise 25 are identical, and that if $\mathfrak{T}$ is the topology of the space $X'$ defined in Exercise 24, then $X''$ can be identified with the space $X'$ carrying the semi-regular topology $\mathfrak{T}^*$ associated with $\mathfrak{T}$ (§ 8, Exercise 20). Furthermore, the set $X''$ can be canonically identified with the set of ultrafilters on $X$. The compact space $X''$ is called the ultrafilter space of $X$.