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Wikipedia / 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. 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

is called if points can be separated from closed sets via (bounded) continuous real-valued functions. In technical terms this means: for any closed set and any point there exists a real-valued continuous function be a bounded function.)

A topological space is called a (alternatively: , or , or ) 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 is 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:

  • 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.

Real-valued continuous functions

denote the family of real-valued continuous functions be the subset of bounded real-valued continuous functions.

In particular:

  • is completely regular if and only if it has the initial topology

  • is completely regular if and only if every closed set can be written as the intersection of a family of zero sets).

  • is completely regular if and only if the cozero sets form a basis

Then ρ will be the finest This construction is universal In the language of category theory, the functor 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.

The category of realcompact 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 is homeomorphic

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:

  1. A topological space is Tychonoff if and only if it can be embedded in a Tychonoff cube.

Compactifications

is dense these are called Hausdorff compactifications the closure 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 there is a unique is the composition

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 is uniformizable. A topological space admits a separated uniform structure if and only if it is Tychonoff.

However, there will always be a finest compatible uniformity, called the fine uniformity becomes the completion

See also

Citations

Bibliography