Type of regular Hausdorff 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. A Tychonoff space is any completely regular space that is also a Hausdorff space; there exist completely regular spaces that are not Tychonoff (i.e. not Hausdorff).
Paul Urysohn had used the notion of completely regular space in a 1925 paper without giving it a name. But it was Andrey Tychonoff who introduced the terminology "completely regular" in 1930.
Definitions.
A topological space formula_1 is called completely regular if points can be separated from closed sets via (bounded) continuous real-valued functions. In technical terms this means: for any closed set formula_2 and any point formula_3 there exists a real-valued continuous function formula_4 such that formula_5 and formula_6 (Equivalently one can choose any two values instead of formula_7 and formula_8 and even require that formula_9 be a bounded function.)
A topological space is called a Tychonoff space (alternatively: T3½ space, or Tπ space, or completely T3 space) if 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.
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:
There are regular Hausdorff spaces that are not completely regular, but such examples are complicated to construct. One of them is the so-called "Tychonoff corkscrew", which contains two points such that any continuous real-valued function on the space has the same value at these two points. An even more complicated construction starts with the Tychonoff corkscrew and builds a regular Hausdorff space called "Hewitt's condensed corkscrew", which is not completely regular in a stronger way, namely, every continuous real-valued function on the space is constant.
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:
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.
Real-valued continuous functions.
For any topological space formula_10 let formula_11 denote the family of real-valued continuous functions on formula_1 and let formula_13 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 formula_11 or formula_15 In particular:
Given an arbitrary topological space formula_25 there is a universal way of associating a completely regular space with formula_26 Let ρ be the initial topology on formula_1 induced by formula_28 or, equivalently, the topology generated by the basis of cozero sets in formula_26 Then ρ will be the finest completely regular topology on formula_1 that is coarser than formula_31 This construction is universal in the sense that any continuous function
formula_32
to a completely regular space formula_33 will be continuous on formula_34 In the language of category theory, the functor that sends formula_25 to formula_36 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 formula_37 in the above construction so that the rings formula_11 and formula_13 are typically only studied for completely regular spaces formula_24
The category of realcompact Tychonoff spaces is anti-equivalent to the category of the rings formula_11 (where formula_1 is realcompact) together with ring homomorphisms as maps. For example one can reconstruct formula_1 from formula_11 when formula_1 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 formula_10 there exists a compact Hausdorff space formula_47 such that formula_1 is homeomorphic to a subspace of formula_49
In fact, one can always choose formula_47 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 formula_1 is dense in formula_52 these are called Hausdorff compactifications of formula_24
Given any embedding of a Tychonoff space formula_1 in a compact Hausdorff space formula_47 the closure of the image of formula_1 in formula_47 is a compactification of formula_24
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 formula_59
It is characterized by the universal property that, given a continuous map formula_9 from formula_1 to any other compact Hausdorff space formula_62 there is a unique continuous map formula_63 that extends formula_9 in the sense that formula_9 is the composition of formula_66 and formula_67
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 formula_1 is uniformizable. A topological space admits a separated uniform structure if and only if it is Tychonoff.
Given a completely regular space formula_1 there is usually more than one uniformity on formula_1 that is compatible with the topology of formula_24 However, there will always be a finest compatible uniformity, called the fine uniformity on formula_24 If formula_1 is Tychonoff, then the uniform structure can be chosen so that formula_74 becomes the completion of the uniform space formula_24
Citations.
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Bibliography.
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