Exercise 27, § 9

; now prove that the canonical mapping $\psi : X'' \to \tilde{X}$ is surjective and that $\tilde{X}$ is compact. Every continuous mapping $f$ of X into a compact space Y can therefore be written in one and only one way $f = h \circ \psi \circ \varphi$, where $h : \tilde{X} \to Y$ is continuous.

27) If X is a discrete space, now prove that the spaces $X''$ and $\tilde{X}$ defined in Exercise 26 are identical, and that if $\mathfrak{T}$ is the topology of the space $X'$ defined in Exercise 25, the space $X''$ is identified with the space $X'$ endowed with the semi-regular topology $\mathfrak{T}^*$ associated with $\mathfrak{T}$ (I, p. 103, Exercise 20). Moreover, the set $X''$ is identified canonically with the set of ultrafilters on X, and one says that the compact space $X''$ is the space of ultrafilters of X.