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 For any topological space X, let C(X) denote the family of real-valued
continuous functions on X and let 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 C(X) or C_b(X). In particular: • A space X is completely regular if and only if it has the
initial topology induced by C(X) or C_b(X). • A space X is completely regular if and only if every closed set can be written as the intersection of a family of
zero sets in X (i.e. the zero sets form a basis for the closed sets of X). • A space X is completely regular if and only if the
cozero sets of X form a
basis for the topology of X. Given an arbitrary topological space (X, \tau) there is a universal way of associating a completely regular space with (X, \tau). Let ρ be the initial topology on X induced by C_{\tau}(X) or, equivalently, the topology generated by the basis of cozero sets in (X, \tau). Then ρ will be the
finest completely regular topology on X that is coarser than \tau. This construction is
universal in the sense that any continuous function f : (X, \tau) \to Y to a completely regular space Y will be continuous on (X, \rho). In the language of
category theory, the
functor that sends (X, \tau) to (X, \rho) 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 C_{\tau}(X) = C_{\rho}(X) in the above construction so that the rings C(X) and C_b(X) are typically only studied for completely regular spaces X. The category of
realcompact Tychonoff spaces is anti-equivalent to the category of the rings C(X) (where X is realcompact) together with ring homomorphisms as maps. For example one can reconstruct X from C(X) when 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.
Embeddings Tychonoff spaces are precisely those spaces that can be
embedded in
compact Hausdorff spaces. More precisely, for every Tychonoff space X, there exists a compact Hausdorff space K such that X is
homeomorphic to a subspace of K. In fact, one can always choose 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.
Compactifications Of particular interest are those embeddings where the image of X is
dense in K; these are called Hausdorff
compactifications of X. Given any embedding of a Tychonoff space X in a compact Hausdorff space K the
closure of the image of X in K is a compactification of X. 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 \beta X. It is characterized by the
universal property that, given a continuous map f from X to any other compact Hausdorff space Y, there is a
unique continuous map g : \beta X \to Y that extends f in the sense that f is the
composition of g and j.
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 X is
uniformizable. A topological space admits a separated uniform structure if and only if it is Tychonoff. Given a completely regular space X there is usually more than one uniformity on X that is compatible with the topology of X. However, there will always be a finest compatible uniformity, called the
fine uniformity on X. If X is Tychonoff, then the uniform structure can be chosen so that \beta X becomes the
completion of the uniform space X. ==See also==