Let $X$ - Tychonoff topological space. Show that Stone–Čech compactification of $X$ can be obtained by taking the closure of the image of the space $X$ under the mapping $\Delta_{f\in C(X, I)}f$ in the space $\prod_{f\in C(X, I)}I_f$, where
$C(X, I)$ is the family af all continous mappings from $X$ to $I=[0, 1],$
$I_f = I = [0, 1]$ for $f\in C(X, I),$
$\prod_{f\in C(X, I)}I_f$ is Tychonoff cube i.e. Cartesian product of unit intervals $I$ by indexing set $C(X, I)$ with product topology,
$\Delta_{f\in C(X, I)}f$ is diagonal mapping i.e
$$\Delta_{f\in C(X, I)}f: X\to \prod_{f\in C(X, I)}I_f \;$$ $$\Delta_{f\in C(X, I)}f: x\mapsto \{f(x)\}_{f\in C(X, I)}\in\prod_{f\in C(X, I)}I_f \; .$$
- Stone–Čech compactification of space $X$ is the largest element in partial order of compactifications of space $X$. This partial order can be defined the following way. Let $Y$, $Z$ - compactifications of $X$ with homeomorphism embeddings $c_Y$ and $c_Z$ respectively. By definition $Z \leq Y$ if there exists continuous mapping $f: Y \to Z$ such that $fc_Y = c_Z$.
I will be glad for any idea, comment, hint or advice.
Some observations and ideas:
Let $e_X: X \to \prod_{f \in C(X,I)} I_f$ be the "diagonal embedding" and denote by $\beta(X)$, as suggested, the space $\overline{e_X[X]}$ in its subspace topology. This is clearly a Hausdorff compactification of $X$ and let's suppose that $(Y,c_Y)$ is some other Hausdorff compactification of $X$ such that $(\beta(X), e_X) \le (Y, c_Y)$ so that we have a continuous $g: Y \to \beta(X)$ such that
$$\forall x \in X: g(c_Y(x)) = e_X(x)\tag{1}$$
It's quite clear that $g$ maps $Y$ onto $\beta(X)$ as $e_X[X] \subseteq g[Y]$ and the former set is dense in $\beta(X)$ by definition.
Also for any $f \in C(X,I)$ we have that $\pi_f \circ g \circ c_Y = f$ on $X$. This implies that $g$ is injective (think about why!) and so $g$ is a homeomorphism between $Y$ and $\beta(X)$. So $\beta(X)$ is maximal.