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).

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

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 real-valued continuous function ${\displaystyle f:X\to \mathbb {R} }$ 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 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.

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 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, 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, which is not completely regular in a stronger way, namely, every continuous real-valued function on the space is constant.