Tychonoff space

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
Separation axioms
in topological spaces
Kolmogorov classification
T0 (Kolmogorov)
T1 (Fréchet)
T2 (Hausdorff)
T2½(Urysohn)
completely T2 (completely Hausdorff)
T3 (regular Hausdorff)
T(Tychonoff)
T4 (normal Hausdorff)
T5 (completely normal
 Hausdorff)
T6 (perfectly normal
 Hausdorff)

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 refers to any completely regular space that is also a Hausdorff space; there exist completely regular spaces that are not Tychonoff (i.e. not Hausdorff).

Tychonoff spaces are named after Andrey Nikolayevich Tychonoff, whose Russian name (Тихонов) is variously rendered as "Tychonov", "Tikhonov", "Tihonov", "Tichonov", etc. who introduced them in 1930 in order to avoid the pathological situation of Hausdorff spaces whose only continuous real-valued functions are constant maps.[1]

Definitions[edit]

Separation of a point from a closed set via a continuous function.

A topological space 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 and any point there exists a real-valued continuous function such that and (Equivalently one can choose any two values instead of and and even demand that be a bounded function.)

A topological space is called a Tychonoff space (alternatively: T 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[edit]

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[edit]

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[edit]

Preservation[edit]

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 bug-eyed line. There are closed quotients of the Moore plane that provide counterexamples.

Real-valued continuous functions[edit]

For any topological space let denote the family of real-valued continuous functions on and let 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 or In particular:

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

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

to a completely regular space will be continuous on In the language of category theory, the functor that sends to is left adjoint to the inclusion functor CRegTop. 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 in the above construction so that the rings and are typically only studied for completely regular spaces

The category of realcompact Tychonoff spaces is anti-equivalent to the category of the rings (where is realcompact) together with ring homomorphisms as maps. For example one can reconstruct from when 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[edit]

Tychonoff spaces are precisely those spaces that can be embedded in compact Hausdorff spaces. More precisely, for every Tychonoff space there exists a compact Hausdorff space such that is homeomorphic to a subspace of

In fact, one can always choose 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[edit]

Of particular interest are those embeddings where the image of is dense in these are called Hausdorff compactifications of Given any embedding of a Tychonoff space in a compact Hausdorff space the closure of the image of in is a compactification of 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 It is characterized by the universal property that, given a continuous map from to any other compact Hausdorff space there is a unique continuous map that extends in the sense that is the composition of and

Uniform structures[edit]

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

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

See also[edit]

Citations[edit]

Bibliography[edit]

  • Gillman, Leonard; Jerison, Meyer (1960). Rings of continuous functions. Graduate Texts in Mathematics, No. 43 (Dover reprint ed.). NY: Springer-Verlag. p. xiii. ISBN 978-048681688-3.
  • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
  • Willard, Stephen (1970). General Topology (Dover reprint ed.). Reading, Massachusetts: Addison-Wesley Publishing Company. ISBN 0-486-43479-6.