In Dugundji's book Topology an interesting way to study topological spaces shows up frequerently: embedding them into polytopes which are defined by the author as arbitrary cartesian products of unit interval endowed with the natural euclidean topology. In particular for completely regular spaces we have:
-Any cartesian product of unit intervals is called a parallelotope. In particular let $I^Y$be the set of all continuous maps from Y into I, and $\{I_f\}$ a collection of unit intervals indexed by these functions. Then we denote by $P^Y$ the particular parallelotope given by $P^Y=\prod_{f\in\ I^Y} I_f$.
-Y is completely regular f and only of it can be embedded in a parallelotope. in particular the map $\rho :Y\ \rightarrow\ P^Y$ $y\ \mapsto\ \{f(y)_f \}$ is a homemorphism onto $\rho (Y)$. We have a map that to any point in the space associates the set of its images under all the elements of $I^Y$.
-Given a set of arbitrary cardinality $\mathscr{A}$ we define the metric space $l^2(\mathscr{A})$ as the one having for: Elements: every $x=\{x_n\}\ \in\ \mathbb{R}^{card(\mathscr{A})}$ such that at most countably many $x_n$ are nonnull (hence we may regard an element as a sequence) and $\sum{x_n}^2$ converges. Topology: the one induced by the euclidean metric $d(x,y)=\sqrt{\sum (x_n-y_n)^2}$
-The subspace of $l^2(\mathbb{N}),\ \{\{x_n\}\ \in\ l^2(\mathbb{N}) |x_n|\leq\frac{1}{n}\}$ is called the Hilbert cube and denoted by $I^{\infty}$.
Now for metrizable spaces we have that:
-Y is metrizable if and only if it can be embedded in $l^2(\mathscr{U})$ where $\mathscr{U}$ is a basis decomposable in a countable collection of nbd finite familes. If Y has a countable basis it can be embedded in $I^{\infty}$
Now, we know that any metrizable space is normal and hence completely regular, and actually completely regualr spaces coincide with gauge spaces, so I would expect that $l^2(\mathscr{U})$ could always be embedded in $P^Y$ somehow, and that they show a non trivial connection. Is this true? What can we say about the relation between tese objects, and about how it reflects in the startting space? Finally is there more to this approach? Which is the meaning of this kind of procedures? Maybe we could see this kind of infinite product spaces as gneralized cooordinate systems for our space?
If $X$ is metrisable (and infinite), let $\kappa$ be the minimal cardinality of a dense subset for $X$. (So $\kappa=\aleph_0$ for separable spaces etc.) As this means (for metrisable spaces) that the minimal size of a base is also $\kappa$ and we can embed $X$ into $[0,1]^\kappa$. The latter Tychonoff cube (as is their more usual name) is not metrisable but we can also embed it into $\ell^2(\kappa)$, which is a nice Hilbert space.
For some purposes it's handy to use one universal space for these spaces, for some other another can be more useful. And yes, $\ell^2(\kappa)$ embeds into $[0,1]^\kappa$ too; that follows purely from being another example of a metrisable space of density $\kappa$.
They have no connection except that they are both universal spaces for metrisable spaces of density and weight $\kappa$. The Tychonoff cube is handy because it's compact and shows we have compactifications. We can also show that $X$ embeds into $\ell^2(\kappa)$ as a subset of a convex subset etc., which can be handy in some so-called selection theorems. I wouldn't look at them as some sort of universal coordinate system; they're handy tools in some proofs.