When defining Stone–Čech compactification we take a Tychonoff space $X$, the space $C_b(X)$ of bounded continuous real functions on $X$, define $I_f$ as closed limited intervals containing $f(X)$ for each $f \in C_b(X)$ and the mapping $$\varepsilon_X : \quad x \in X \mapsto (f(x))_{f \in C_b(X)} \prod_{f \in C_b(X)} I_f.$$ My notes assert that this mapping is an embedding by completely regular spaces properties. Defining $\beta X$ as $\beta X = \overline{\varepsilon_X (X)}$ we obtain the Stone–Čech compactificationwith the pair $(\beta X, \varepsilon_X)$.
To have a compactification we need that $X$ is a Hausdorff space, $\beta X$ be a compact Hausdorff space and $\varepsilon_X$ be a homeomorphism between $X$ and a dense subspace of $\beta X$. My questions are:
- $X$ is merely Tychonoff and not Hausdorff, how does this work then?
- How can we assert that $\varepsilon_X$ is homeomorphic to a dense subspace of $\prod I_f$?
We know that it is homeomorphic to its image, but is the image dense? How and why?