Tychonoff space

Complete regularity and the Tychonoff property are well-behaved with respect to initial topologies. Specifically, complete regularity is preserved by taking arbitrary initial topologies and the Tychonoff property is preserved by taking point-separating initial topologies. It follows that:

• Every subspace of a completely regular or Tychonoff space has the same property.
• A nonempty product space is completely regular (respectively Tychonoff) if and only if each factor space is completely regular (respectively Tychonoff).

Like all separation axioms, complete regularity is not preserved by taking final topologies. In particular, quotients of completely regular spaces need not be regular. Quotients of Tychonoff spaces need not even be Hausdorff, with one elementary counterexample being the line with two origins. There are closed quotients of the Moore plane that provide counterexamples.

For any topological space ${\displaystyle X,}$ let ${\displaystyle C(X)}$ denote the family of real-valued continuous functions on ${\displaystyle X}$ and let ${\displaystyle C_{b}(X)}$ be the subset of bounded real-valued continuous functions.

Completely regular spaces can be characterized by the fact that their topology is completely determined by ${\displaystyle C(X)}$ or ${\displaystyle C_{b}(X).}$ In particular:

• A space ${\displaystyle X}$ is completely regular if and only if it has the initial topology induced by ${\displaystyle C(X)}$ or ${\displaystyle C_{b}(X).}$
• A space ${\displaystyle X}$ is completely regular if and only if every closed set can be written as the intersection of a family of zero sets in ${\displaystyle X}$ (i.e. the zero sets form a basis for the closed sets of ${\displaystyle X}$).
• A space ${\displaystyle X}$ is completely regular if and only if the cozero sets of ${\displaystyle X}$ form a basis for the topology of ${\displaystyle X.}$

Given an arbitrary topological space ${\displaystyle (X,\tau )}$ there is a universal way of associating a completely regular space with ${\displaystyle (X,\tau ).}$ Let ρ be the initial topology on ${\displaystyle X}$ induced by ${\displaystyle C_{\tau }(X)}$ or, equivalently, the topology generated by the basis of cozero sets in ${\displaystyle (X,\tau ).}$ Then ρ will be the finest completely regular topology on ${\displaystyle X}$ that is coarser than ${\displaystyle \tau .}$ This construction is universal in the sense that any continuous function

$${\displaystyle f:(X,\tau )\to Y}$$

to a completely regular space ${\displaystyle Y}$ will be continuous on ${\displaystyle (X,\rho ).}$ In the language of category theory, the functor that sends ${\displaystyle (X,\tau )}$ to ${\displaystyle (X,\rho )}$ is left adjoint to the inclusion functor CReg → Top. Thus the category of completely regular 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 ${\displaystyle C_{\tau }(X)=C_{\rho }(X)}$ in the above construction so that the rings ${\displaystyle C(X)}$ and ${\displaystyle C_{b}(X)}$ are typically only studied for completely regular spaces ${\displaystyle X.}$

The category of realcompact Tychonoff spaces is anti-equivalent to the category of the rings ${\displaystyle C(X)}$ (where ${\displaystyle X}$ is realcompact) together with ring homomorphisms as maps. For example one can reconstruct ${\displaystyle X}$ from ${\displaystyle C(X)}$ when ${\displaystyle X}$ 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.

Tychonoff spaces are precisely those spaces that can be embedded in compact Hausdorff spaces. More precisely, for every Tychonoff space ${\displaystyle X,}$ there exists a compact Hausdorff space ${\displaystyle K}$ such that ${\displaystyle X}$ is homeomorphic to a subspace of ${\displaystyle K.}$

In fact, one can always choose ${\displaystyle K}$ 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.

Of particular interest are those embeddings where the image of ${\displaystyle X}$ is dense in ${\displaystyle K;}$ these are called Hausdorff compactifications of ${\displaystyle X.}$ Given any embedding of a Tychonoff space ${\displaystyle X}$ in a compact Hausdorff space ${\displaystyle K}$ the closure of the image of ${\displaystyle X}$ in ${\displaystyle K}$ is a compactification of ${\displaystyle X.}$ In the same 1930 article where Tychonoff defined completely regular spaces, he also proved that every Tychonoff space has a Hausdorff compactification.[2]

Among those Hausdorff compactifications, there is a unique "most general" one, the Stone–Čech compactification ${\displaystyle \beta X.}$ It is characterized by the universal property that, given a continuous map ${\displaystyle f}$ from ${\displaystyle X}$ to any other compact Hausdorff space ${\displaystyle Y,}$ there is a unique continuous map ${\displaystyle g:\beta X\to Y}$ that extends ${\displaystyle f}$ in the sense that ${\displaystyle f}$ is the composition of ${\displaystyle g}$ and ${\displaystyle j.}$

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 ${\displaystyle X}$ is uniformizable. A topological space admits a separated uniform structure if and only if it is Tychonoff.

Given a completely regular space ${\displaystyle X}$ there is usually more than one uniformity on ${\displaystyle X}$ that is compatible with the topology of ${\displaystyle X.}$ However, there will always be a finest compatible uniformity, called the fine uniformity on ${\displaystyle X.}$ If ${\displaystyle X}$ is Tychonoff, then the uniform structure can be chosen so that ${\displaystyle \beta X}$ becomes the completion of the uniform space ${\displaystyle X.}$