Tychonoff space

Summary

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).

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)

Paul Urysohn had used the notion of completely regular space in a 1925 paper[1] without giving it a name. But it was Andrey Tychonoff who introduced the terminology completely regular in 1930.[2]

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 require 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 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:

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,[3][4] 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,[5][6] which is not completely regular in a stronger way, namely, every continuous real-valued function on the space is constant.

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 line with two origins. 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[2] 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 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

  • Stone–Čech compactification – a universal map from a topological space X to a compact Hausdorff space βX, such that any map from X to a compact Hausdorff space factors through βX uniquely; if X is Tychonoff, then X is a dense subspace of βX

Citations edit

  1. ^ Urysohn, Paul (1925). "Über die Mächtigkeit der zusammenhängenden Mengen". Mathematische Annalen. 94 (1): 262–295. doi:10.1007/BF01208659. See pages 291 and 292.
  2. ^ a b Tychonoff, A. (1930). "Über die topologische Erweiterung von Räumen". Mathematische Annalen. 102 (1): 544–561. doi:10.1007/BF01782364.
  3. ^ Willard 1970, Problem 18G.
  4. ^ Steen & Seebach 1995, Example 90.
  5. ^ Steen & Seebach 1995, Example 92.
  6. ^ Hewitt, Edwin (1946). "On Two Problems of Urysohn". Annals of Mathematics. 47 (3): 503–509. doi:10.2307/1969089.

Bibliography edit