spaces CReg is a reflective subcategory of Top, the category of topological spaces. By taking Kolmogorov quotients, one sees that the subcategory of Tychonoff spaces is also reflective. One can show that in the above construction so that the rings and are typically only studied for completely regular spaces The category of realcompact Tychonoff spaces is anti-equivalent to the category of the rings (where is realcompact) together with ring homomorphisms as maps. For example one can reconstruct from when is (real) compact. The algebraic theory of these rings is therefore subject of intensive studies. A vast generalization of this class of rings that still resembles many properties of Tychonoff spaces, but is also applicable in real algebraic geometry, is the class of real closed rings. Embeddings Tychonoff spaces are precisely those spaces that can be embedded in compact Hausdorff spaces. More precisely, for every Tychonoff space there exists a compact Hausdorff space such that is homeomorphic to a subspace of In fact, one can always choose to be a Tychonoff cube (i.e. a possibly infinite product of unit intervals). Every Tychonoff cube is compact Hausdorff as a consequence of Tychonoff's theorem. Since every subspace of a compact Hausdorff space is Tychonoff one has: A topological space is Tychonoff if and only if it can be embedded in a Tychonoff cube. Compactifications Of particular interest are those embeddings where the image of is dense in these are called Hausdorff compactifications of Given any embedding of a Tychonoff space in a compact Hausdorff space the closure of the image of in is a compactification of In the same 1930 article where Tychonoff defined completely regular spaces, he also proved that every Tychonoff space has a Hausdorff compactification. Among those Hausdorff compactifications, there is a unique "most general" one, the Stone–Čech compactification It is characterized by the universal property that, given a continuous map from to any other compact Hausdorff space there is a unique continuous map that extends in the sense that is the composition of and Uniform structures Complete regularity is exactly the condition necessary for the existence of uniform structures on a topological space. In other words, every uniform space has a completely regular topology and every completely regular space is uniformizable. A topological space admits a separated uniform structure if and only if it is Tychonoff. Given a completely regular space there is usually more than one uniformity on