In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. These conditions are examples of separation axioms. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... Euclid, detail from The School of Athens by Raphael. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. ...

Tychonoff spaces are named after Andrey Nikolayevich Tychonoff, whose Russian name (Тихонов) is also sometimes transliterated as "Tychonov", "Tikhonov", "Tihonov", or "Tichonov". Andrey Nikolayevich Tychonoff (Андрей Николаевич Тихонов: October 30, 1906–1993) was a Russian mathematician. ...

Definitions

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 0 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. ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P... In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... Point can refer to: Look up Point in Wiktionary, the free dictionary // Mathematics In mathematics: Point (geometry), an entity that has a location in space but no extent Fixed point (mathematics), a point that is mapped to itself by a mathematical function Point at infinity Point group Point charge, an... In predicate logic, an existential quantification is the predication of a property or relation to at least one member of the domain. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, the real line is simply the set of real numbers. ... 0 (zero), alternatively called naught, nil, nada, ought, zilch, zip, nothing or nought, is both a number and a numeral. ... Look up one in Wiktionary, the free dictionary. ... In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way. ...

X is a Tychonoff space, or T_3½ space, or T_π space, or completely T₃ space if and only if it is both completely regular and Hausdorff). In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...

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. In general topology, the separation axioms have had a convoluted history, with many competing meanings for the same term, and many competing terms for the same concept. ...

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. In topology and related branches of mathematics, the T0 spaces or Kolmogorov spaces form a broad class of well behaved topological spaces. ... In topology and related branches of mathematics, the T0 spaces or Kolmogorov spaces form a broad class of well behaved topological spaces. ... In topology and related branches of mathematics, the T0 spaces or Kolmogorov spaces form a broad class of well behaved topological spaces. ...

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: Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ... In mathematics, the real line is simply the set of real numbers. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...

• 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.
• The Niemytzki plane is an example of a Tychonoff space which is not normal.

In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In mathematics, a metric space is a set (or space) where a distance between points is defined. ... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In topology and related fields of mathematics, regular spaces and T3 spaces are particularly nice kinds of topological spaces. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, a total order or linear order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... In mathematics, the order topology is a topology that can be defined on any totally ordered set. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... In the mathematical field of topology, a uniform space is a set with a uniform structure. ... In topology, a CW complex is a type of topological space introduced by J.H.C. Whitehead to meet the needs of homotopy theory. ... In topology and related branches of mathematics, normal spaces, T4 spaces, and T5 spaces are particularly nice kinds of topological spaces. ... In topology and related branches of mathematics, normal spaces, T4 spaces, and T5 spaces are particularly nice kinds of topological spaces. ...

Properties

Complete regularity and Tychonoff-ness are preserved by taking initial topologies. In particular, all subspaces and product spaces of Tychonoff or completely regular spaces have the same property. In topology and related areas of mathematics, the initial topology (projective topology or weak topology) on a set , with respect to a family of functions on , is the coarsest topology on X which makes those functions continuous. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ...

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. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... In mathematics, the term dense has at least three different meanings. ... In mathematics, compactification is applied to topological spaces to make them compact spaces. ...

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. In mathematics, compactification is applied to topological spaces to make them compact spaces. ... In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ... In predicate logic and technical fields that depend on it, uniqueness quantification, or unique existential quantification, is an attempt to formalise the notion of something being true for exactly one thing, or exactly one thing of a certain type. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...

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. In the mathematical field of topology, a uniform space is a set with a uniform structure. ... In topology, one defines uniform spaces in order to study concepts such as uniform continuity, completeness and uniform convergence. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For...