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Suppose that X is a topological space.
X is a completely regular space iff, given any closed set F and any point x that does not belong to F, there is a continuous function f from X to the real line R such that f(x) is and f(y) is 1 for every y in F. In fancier terms, this condition says that x and F can be separated by a function.
X is a Tychonoff space, or T3½ space, or Tπ space, or completely T3 space if and only if it is both completely regular and Hausdorff).
Note that some mathematical literature uses different definitions for the term "completely regular" and the terms involving "T". The definitions that we have given here are the ones usually used today; however, some authors switch the meanings of the two kinds of terms, or use all terms synonymously for only one condition. In Wikipedia, we will use the terms "completely regular" and "Tychonoff" freely, but we'll avoid the less clear "T" terms. In other literature, you should take care to find out which definitions the author is using. (The phrase "completely regular Hausdorff", however, is unambiguous, and always means a Tychonoff space.) For more on this issue, see History of the separation axioms.
Completely regular spaces and Tychonoff spaces are related through the notion of Kolmogorov equivalence. A topological space is Tychonoff iff it's both completely regular and T0. On the other hand, a space is completely regular iff its Kolmogorov quotient is Tychonoff.
Examples and counterexamples
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:
- 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.
- Generalising both the metric spaces and the topological groups, every uniform space is completely regular.
- Every CW complex is Tychonoff.
- Every normal regular space is completely regular, and every normal Hausdorff space is Tychonoff.
Tychonoff spaces are precisely those topological spaces which can be embedded in a compact Hausdorff space. More precisely, for every Tychonoff space X, there exists a compact Hausdorff space K and an injective continuous map j from X to K such that the inverse of j is also continuous. Of particular interest are those embeddings where j(X) is dense in K; these are called Hausdorff compactifications of X.
Among those Hausdorff compactifications, there is a unique "most general" one, the Stone-Cech compactification βX. It is characterised 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 from βX to Y that extends f in the sense that f is the composition of g and j.
As mentioned above, every uniform space has a completely regular topology. Conversely, any completely regular space X can be made into a uniform space in some way. If X is Tychonoff, then the uniform structure can be chosen so that βX becomes the completion of the uniform space X.
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