Suppose that X is a topological space.
X is a completely regular space if given any closed set F and any point x that does not belong to F, then there is a continuous function f from X to the real line R such that f(x) is 0 and, for every y in F, f(y) is 1. In other terms, this condition says that x and F can be separated by a continuous function.
X is a Tychonoff space, or T3½ space, or Tπ space, or completely T3 space if it is both completely regular and Hausdorff.
Note that some mathematical literature uses different definitions for the term "completely regular" and the terms involving "T". The definitions that we have given here are the ones usually used today; however, some authors switch the meanings of the two kinds of terms, or use all terms synonymously for only one condition. In Wikipedia, we will use the terms "completely regular" and "Tychonoff" freely, but we'll avoid the less clear "T" terms. In other literature, you should take care to find out which definitions the author is using. (The phrase "completely regular Hausdorff", however, is unambiguous, and always means a Tychonoff space.) For more on this issue, see History of the separation axioms.
Completely regular spaces and Tychonoff spaces are related through the notion of Kolmogorov equivalence. A topological space is Tychonoff if and only if it's both completely regular and T0. On the other hand, a space is completely regular if and only if its Kolmogorov quotient is Tychonoff.
Examples and counterexamples
Almost every topological space studied in mathematical analysis is Tychonoff, or at least completely regular. For example, the real line is Tychonoff under the standard Euclidean topology. Other examples include:
- Every metric space is Tychonoff; every pseudometric space is completely regular.
- Every locally compact regular space is completely regular, and therefore every locally compact Hausdorff space is Tychonoff.
- In particular, every topological manifold is Tychonoff.
- Every totally ordered set with the order topology is Tychonoff.
- Every topological group is completely regular.
- Generalising both the metric spaces and the topological groups, every uniform space is completely regular. The converse is also true: every completely regular space is uniformisable.
- Every CW complex is Tychonoff.
- Every normal regular space is completely regular, and every normal Hausdorff space is Tychonoff.
- The Niemytzki plane is an example of a Tychonoff space which is not normal.
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 (resp. Tychonoff) if and only if each factor space is completely regular (resp. 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. There are closed quotients of the Moore plane which provide counterexamples.
Real-valued continuous functions
Completely regular spaces can be characterized by the fact that their topology is completely determined by C(X) or C*(X). In particular:
- A space X is completely regular if and only if it has the initial topology induced by C(X) or C*(X).
- A space X is completely regular if and only if every closed set can be written as the intersection of a family of zero sets in X (i.e. the zero sets form a basis for the closed sets of X).
- A space X is completely regular if and only if the cozero sets of X form a basis for the topology of X.
Given an arbitrary topological space (X, τ) there is a universal way of associating a completely regular space with (X, τ). Let ρ be the initial topology on X induced by Cτ(X) or, equivalently, the topology generated by the basis of cozero sets in (X, τ). Then ρ will be the finest completely regular topology on X which is coarser than τ. This construction is universal in the sense that any continuous function
to a completely regular space Y will be continuous on (X, ρ). In the language of category theory, the functor which sends (X, τ) to (X, ρ) 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 Cτ(X) = Cρ(X) in the above construction so that the rings C(X) and C*(X) are typically only studied for completely regular spaces X.
The category of real compact Tychonoff spaces is anti-equivalent to the category of the rings C(X) (where X is real compact) together with ring homomorphisms as maps. For example one can reconstruct $X$ from C(X) when X is (real) compact. The algebraic theory of these rings is therefore subject of intensive studies. A vast generalisation of this class of rings which 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 which can be embedded in compact Hausdorff spaces. More precisely, for every Tychonoff space X, there exists a compact Hausdorff space K such that X is homeomorphic to a subspace of K.
In fact, one can always choose 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 X is dense in K; these are called Hausdorff compactifications of X. Given any embedding of a Tychonoff space X in a compact Hausdorff space K the closure of the image of X in K is a compactification of X.
Among those Hausdorff compactifications, there is a unique "most general" one, the Stone–Čech compactification βX. It is characterised by the universal property that, given a continuous map f from X to any other compact Hausdorff space Y, there is a unique continuous map g from βX to Y that extends f in the sense that f is the composition of g and 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 X is uniformizable. A topological space admits a separated uniform structure if and only if it is Tychonoff.
Given a completely regular space X there is usually more than one uniformity on X that is compatible with the topology of X. However, there will always be a finest compatible uniformity, called the fine uniformity on X. If X is Tychonoff, then the uniform structure can be chosen so that βX becomes the completion of the uniform space X.