Embedding Theorem for topological spaces, and in general

Products of topological spaces give us nice embedding characterizations of spaces. Let $X$ be a topological space. Then we say that a topological space $Y$ is $X$-compact if $Y$ is isomorphic to a closed subspace of some product $X^{I}$, so the products $X^{I}$ are in some sense the universal $X$-compact spaces. We say that a space $Y$ is $X$-completely regular if $Y$ is isomorphic to a subspace of some product $X^{I}$. Some nice topological properties may be characterized using $X$-compactness and $X$-complete regularity.

For instance, the $[0,1]$-completely regular spaces are simply the completely regular spaces. The $\{0,1\}$-completely regular spaces are the zero-dimensional spaces. A topological space is compact and Hausdorff if and only if it is $[0,1]$-compact. A topological space is a Boolean space(i.e. compact and zero-dimensional) if and only if it is $\{0,1\}$-compact. A topological space whose cardinality is below the first measurable cardinal is realcompact ($\mathbb{R}$-compact spaces are called realcompact spaces) if and only if it can be given a compatible complete uniformity. Furthermore, a topological space whose cardinality is below the first measurable cardinal is $\mathbb{N}$-compact if and only if it can be given a compatible complete non-Archimedean uniformity (a non-Archimedean uniform space is a uniform space generated by equivalence relations). I must mention that the restriction that the cardinality of your space is below the first measurable cardinal is a very minor restriction. The existence of a measurable cardinal goes beyond the standard axioms of set theory. Furthermore, measurable cardinals are extremely large if they do exist.

On a related note, every metric space can be isometrically embedded into some space $\ell^{\infty}(A)$ for some set $A$ (See Isbells book on Uniform Space for a proof). Therefore the spaces $\ell^{\infty}(A)$ are in a sense the universal metric spaces. In particular, since every metric space is uniformly homeomorphic to a bounded metric space, every metric space is uniformly homeomorphic to a subspace of the unit ball of $\ell^{\infty}(A)$. Since every uniform space is induced by a family of pseudometrics, we conclude that every uniform space is uniformly homeomorphic to a subspace of a product of the unit balls in some $\ell^{\infty}(A)$. In particular, a topological space is can be given a compatible complete uniformity if and only if it is isomorphic to some closed subspace of some product $\ell^{\infty}(A)^{I}$.

Every $T_0$ space is, in a canonical way, a subspace of an algebraic lattice equipped with its Scott topology. Some details can be found in this nLab article.

I'm not quite sure what to do with the general question. Some possible inspiration might come from the study of Chu constructions -- see for examples these notes by Vaughan Pratt. Many familiar mathematical objects can be realized concretely as Chu spaces in one way or another.