I read about various constructions of Stone-Čech compactification on Wikipedia. Regarding the construction using unit interval, I have a question about checking the universal property.
"In fact, this closure is the Stone–Čech compactification. To verify this, we just need to verify that the closure satisfies the appropriate universal property. We do this first for $K = [0, 1]$, where the desired extension of $f : X → [0, 1]$ is just the projection onto the f coordinate in $[0, 1]^C$.
In order to then get this for general compact Hausdorff K we use the above to note that K can be embedded in some cube, extend each of the coordinate functions and then take the product of these extension."
I dont understand the statement in bold letters. Is that "some cube" just a product of closed unit intervals? And how exactly do we extend the coordinate functions? Each of the functions goes from X to $[0, 1]$, so if we want to embed the $[0, 1]$ in "some cube", say of dimension n, we just make n copies of image of each function in $[0, 1]$?
Thank you for your help.
So we see $X$ as a homeomorph of $e_X[X]$ where $e_X: X \to C:= [0,1]^{C(X,I)}$ (defined by $\pi_f \circ e_X = f$ for all $f \in C(X,I)$) is the canonical embedding of the Tychonoff space $X$ into the cube and $\beta X $ is then defined as its closure $\overline{e_X[X]}^{(C)}$ in that cube.
Now, if $K$ is any compact Hausdorff space we can in the same way, see it as a subspace of a cube $C' = [0,1]^{C(K,I)}$ via its own embedding $e_K$. Given $f: X \to K$ we want to extend $f$ to $\beta f : \beta X \to K$ so that $\beta f \circ e_X = f$ on $X$. For each $f' \in C(K,I)$ we have that $f' \circ f \in C(X,I)$ and so this $f' \circ f$ is a "coordinate" of the cube $C$ and so we can define a map $F: C \to C'$ by $\pi_{f'} \circ F = \pi_{f' \circ f}$ and this is uniquely determined and continuous (by standard facts on product maps; this $F$ is the "product" Wikipedia talks about). It's then standard to verify that $e_X[X]$ (and so its closure too, by continuity) is mapped into $e_K[K]\simeq K$ by this $F$ and so we can use $(e_K)^{-1} \circ F\restriction_{\beta(X)}$ as the promised extension $\beta X \to K$.