Tychonoff space

In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms.

Separation axioms
in topological spaces
Kolmogorov classification
T0 (Kolmogorov)
T1 (Fréchet)
T2 (Hausdorff)
T2½(Urysohn)
completely T2 (completely Hausdorff)
T3 (regular Hausdorff)
T3½(Tychonoff)
T4 (normal Hausdorff)
T5 (completely normal
Hausdorff)
T6 (perfectly normal
Hausdorff)

Tychonoff spaces are named after Andrey Nikolayevich Tychonoff, whose Russian name (Тихонов) is variously rendered as "Tychonov", "Tikhonov", "Tihonov", "Tichonov" etc.

Definitions

A topological space, ${\displaystyle X}$, is called completely regular exactly in case points can be separated from closed sets via (bounded) continuous real-valued functions. In technical terms this means: for any closed set ${\displaystyle A\subseteq X}$ and any point ${\displaystyle x\in X\setminus A}$, then there exists a real-valued continuous function ${\displaystyle f:X\longrightarrow \mathbb {R} }$ such that ${\displaystyle f(x)=1}$ and ${\displaystyle f\mid A\equiv 0}$. (Equivalently one can choose any two values instead of ${\displaystyle 0}$ and ${\displaystyle 1}$ and even demand that ${\displaystyle f}$ be a bounded function.)

A topological space, ${\displaystyle X}$, is furthermore called a Tychonoff space (alternatively: T space, or Tπ space, or completely T3 space) in case it is a completely regular Hausdorff space.

Remark. 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.

Naming conventions

Across mathematical literature different conventions are applied when it comes to the term "completely regular" and the "T"-Axioms. The definitions in this section are in typical modern usage. Some authors, however, switch the meanings of the two kinds of terms, or use all terms interchangeably. In Wikipedia, the terms "completely regular" and "Tychonoff" are used freely and the "T"-notation is generally avoided. In standard literature, caution is thus advised, to find out which definitions the author is using. For more on this issue, see History of the separation axioms.

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:

Properties

Preservation

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. There are closed quotients of the Moore plane that provide counterexamples.

Real-valued continuous functions

For any topological space X, let C(X) denote the family of real-valued continuous functions on X and let Cb(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 C(X) or Cb(X). In particular:

• A space X is completely regular if and only if it has the initial topology induced by C(X) or Cb(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 that is coarser than τ. This construction is universal in the sense that any continuous function

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

to a completely regular space Y will be continuous on (X, ρ). In the language of category theory, the functor that 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 Cb(X) are typically only studied for completely regular spaces X.

The category of realcompact Tychonoff spaces is anti-equivalent to the category of the rings C(X) (where X is realcompact) 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 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 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.

Compactifications

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.

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 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.

References

• Stephen Willard, General Topology, (1970) Addison-Wesley Publishing Company, Reading Massachusetts.
• Gillman, Leonard; Jerison, Meyer Rings of continuous functions. Reprint of the 1960 edition. Graduate Texts in Mathematics, No. 43. Springer-Verlag, New York-Heidelberg, 1976. xiii+300 pp