Tychonoff space

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)
History

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).

Definitions

A topological space ${\displaystyle X}$ 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 ${\displaystyle A\subseteq X}$ and any point ${\displaystyle x\in X\setminus A,}$ there exists a

${\displaystyle f:X\to \mathbb {R} }$ such that ${\displaystyle f(x)=1}$ and ${\displaystyle f\vert _{A}=0.}$ (Equivalently one can choose any two values instead of ${\displaystyle 0}$ and ${\displaystyle 1}$ and even require that ${\displaystyle f}$ be a bounded function.)

A topological space is called a Tychonoff space (alternatively: T_3½ space, or T_π 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

is Tychonoff.

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.

Almost every topological space studied in mathematical analysis is Tychonoff, or at least completely regular. For example, the

. Other examples include:

• Every metric space is Tychonoff; every pseudometric space is completely regular.
• Every
  locally compact regular space
  is completely regular, and therefore every locally compact Hausdorff space is Tychonoff.
• In particular, every topological manifold is Tychonoff.
• Every
  totally ordered set with the order topology
  is Tychonoff.
• Every topological group is completely regular.
• Every
  pseudometrizable
  space is completely regular, but not Tychonoff if the space is not Hausdorff.
• Every
  seminormed space is completely regular (both because it is pseudometrizable and because it is a topological vector space
  , hence a topological group). But it will not be Tychonoff if the seminorm is not a norm.
• Generalizing both the metric spaces and the topological groups, every uniform space is completely regular. The converse is also true: every completely regular space is uniformisable.
• Every CW complex is Tychonoff.
• Every normal regular space is completely regular, and every normal Hausdorff space is Tychonoff.
• The
  Niemytzki plane is an example of a Tychonoff space that is not normal
  .

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.

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

that provide counterexamples.

Real-valued continuous functions

For any topological space ${\displaystyle X,}$ let ${\displaystyle C(X)}$ denote the family of real-valued

continuous functions

on ${\displaystyle X}$ and let ${\displaystyle C_{b}(X)}$ 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 ${\displaystyle C(X)}$ or ${\displaystyle C_{b}(X).}$ In particular:

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

Given an arbitrary topological space ${\displaystyle (X,\tau )}$ there is a universal way of associating a completely regular space with ${\displaystyle (X,\tau ).}$ Let ρ be the initial topology on ${\displaystyle X}$ induced by ${\displaystyle C_{\tau }(X)}$ or, equivalently, the topology generated by the basis of cozero sets in ${\displaystyle (X,\tau ).}$ Then ρ will be the

completely regular topology on ${\displaystyle X}$ that is coarser than ${\displaystyle \tau .}$ This construction is universal in the sense that any continuous function $${\displaystyle f:(X,\tau )\to Y}$$ to a completely regular space ${\displaystyle Y}$ will be continuous on ${\displaystyle (X,\rho ).}$ In the language of category theory, the functor that sends ${\displaystyle (X,\tau )}$ to ${\displaystyle (X,\rho )}$ is , one sees that the subcategory of Tychonoff spaces is also reflective.

One can show that ${\displaystyle C_{\tau }(X)=C_{\rho }(X)}$ in the above construction so that the rings ${\displaystyle C(X)}$ and ${\displaystyle C_{b}(X)}$ are typically only studied for completely regular spaces ${\displaystyle X.}$

The category of realcompact Tychonoff spaces is anti-equivalent to the category of the rings ${\displaystyle C(X)}$ (where ${\displaystyle X}$ is realcompact) together with ring homomorphisms as maps. For example one can reconstruct ${\displaystyle X}$ from ${\displaystyle C(X)}$ when ${\displaystyle X}$ 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.

Tychonoff spaces are precisely those spaces that can be

. More precisely, for every Tychonoff space ${\displaystyle X,}$ there exists a compact Hausdorff space ${\displaystyle K}$ such that ${\displaystyle X}$ is to a subspace of ${\displaystyle K.}$

In fact, one can always choose ${\displaystyle K}$ 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.

Of particular interest are those embeddings where the image of ${\displaystyle X}$ is

in ${\displaystyle K;}$ these are called Hausdorff compactifications of ${\displaystyle X.}$ Given any embedding of a Tychonoff space ${\displaystyle X}$ in a compact Hausdorff space ${\displaystyle K}$ the closure of the image of ${\displaystyle X}$ in ${\displaystyle K}$ is a compactification of ${\displaystyle X.}$ 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 ${\displaystyle \beta X.}$ It is characterized by the universal property that, given a continuous map ${\displaystyle f}$ from ${\displaystyle X}$ to any other compact Hausdorff space ${\displaystyle Y,}$ there is a

continuous map ${\displaystyle g:\beta X\to Y}$ that extends ${\displaystyle f}$ in the sense that ${\displaystyle f}$ is the of ${\displaystyle g}$ and ${\displaystyle j.}$

Complete regularity is exactly the condition necessary for the existence of

has a completely regular topology and every completely regular space ${\displaystyle X}$ is

uniformizable

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

Given a completely regular space ${\displaystyle X}$ there is usually more than one uniformity on ${\displaystyle X}$ that is compatible with the topology of ${\displaystyle X.}$ However, there will always be a finest compatible uniformity, called the

fine uniformity

on ${\displaystyle X.}$ If ${\displaystyle X}$ is Tychonoff, then the uniform structure can be chosen so that ${\displaystyle \beta X}$ becomes the

completion

of the uniform space ${\displaystyle X.}$

See also

• Stone–Čech compactification – Concept in topology

Bibliography

• ISBN 978-048681688-3
  .
• MR 0507446
  .
• Willard, Stephen (1970). General Topology (Dover reprint ed.). Reading, Massachusetts: Addison-Wesley Publishing Company.
  ISBN 0-486-43479-6
  .