Tychonoff space

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In
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
and related branches of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Tychonoff spaces and completely regular spaces are kinds of
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s. These conditions are examples of
separation axiom 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. Some of these restrictions are given by the separation axioms. These are sometimes ...
s. A Tychonoff space refers to any completely regular space that is also a
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
; there exist completely regular spaces that are not Tychonoff (i.e. not Hausdorff). Tychonoff spaces are named after Andrey Nikolayevich Tychonoff, whose
Russian Russian(s) refers to anything related to Russia, including: *Russians (, ''russkiye''), an ethnic group of the East Slavic peoples, primarily living in Russia and neighboring countries *Rossiyane (), Russian language term for all citizens and peo ...
name (Тихонов) is variously rendered as "Tychonov", "Tikhonov", "Tihonov", "Tichonov", etc. who introduced them in 1930 in order to avoid the pathological situation of
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
s whose only continuous real-valued functions are constant maps.

# Definitions

A topological space $X$ is called if points can be separated from closed sets via (bounded) continuous real-valued functions. In technical terms this means: for any
closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...
$A \subseteq X$ and any
point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...
$x \in X \setminus A,$
there exists In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, w ...
a
real-valued In mathematics, value may refer to several, strongly related notions. In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as or an i ...
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
$f : X \to \R$ such that $f\left(x\right)=1$ and $f\vert_ = 0.$ (Equivalently one can choose any two values instead of $0$ and $1$ and even demand that $f$ be a bounded function.) A topological space is called a (alternatively: , or , or ) if it is a completely regular
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
. Remark. Completely regular spaces and Tychonoff spaces are related through the notion of
Kolmogorov equivalence In topology and related branches of mathematics, a topological space ''X'' is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of ''X'', at least one of them has a Neighbourhood (mathematics), ne ...
. 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 In topology and related branches of mathematics, a topological space ''X'' is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of ''X'', at least one of them has a neighborhood not containing t ...
is Tychonoff.

# Naming conventions

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 The history of the separation axioms in general topology has been convoluted, with many meanings competing for the same terms and many terms competing for the same concept. Origins Before the current general definition of topological space, th ...
.

# Examples and counterexamples

Almost every topological space studied in
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
is Tychonoff, or at least completely regular. For example, the
real line In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
is Tychonoff under the standard
Euclidean topology In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean distance, Euclidean metric. Definition The Euclidean norm on \R^n is the non-negative f ...
. Other examples include: * Every
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
is Tychonoff; every
pseudometric space In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced by Đuro Kurepa in 1934. In the same way as every normed space is a metric ...
is completely regular. * Every locally compact
regular space In topology and related fields of mathematics, a topological space ''X'' is called a regular space if every closed subset ''C'' of ''X'' and a point ''p'' not contained in ''C'' admit non-overlapping open neighborhoods. Thus ''p'' and ''C'' ca ...
is completely regular, and therefore every locally compact Hausdorff space is Tychonoff. * In particular, every
topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathe ...
is Tychonoff. * Every
totally ordered set In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive) ...
with the
order topology In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If ''X'' is a totally ordered set, t ...
is Tychonoff. * Every
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
is completely regular. * Generalizing both the metric spaces and the topological groups, every
uniform space In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifo ...
is completely regular. The converse is also true: every completely regular space is uniformisable. * Every
CW complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...
is Tychonoff. * Every
normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
regular space is completely regular, and every normal Hausdorff space is Tychonoff. * The
Niemytzki plane In mathematics, the Moore plane, also sometimes called Niemytzki plane (or Nemytskii plane, Nemytskii's tangent disk topology), is a topological space. It is a completely regular Hausdorff space (also called Tychonoff space) that is not normal. I ...
is an example of a Tychonoff space that is not
normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
.

# Properties

## 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 In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemi ...
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 bug-eyed line. There are closed quotients of the
Moore plane In mathematics, the Moore plane, also sometimes called Niemytzki plane (or Nemytskii plane, Nemytskii's tangent disk topology), is a topological space. It is a completely regular Hausdorff space (also called Tychonoff space) that is not normal. I ...
that provide counterexamples.

## Real-valued continuous functions

For any topological space $X,$ let $C\left(X\right)$ denote the family of real-valued
continuous functions In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
on $X$ and let $C_b\left(X\right)$ 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 $C\left(X\right)$ or $C_b\left(X\right).$ In particular: * A space $X$ is completely regular if and only if it has the
initial topology In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on ''X'' tha ...
induced by $C\left(X\right)$ or $C_b\left(X\right).$ * A space $X$ is completely regular if and only if every closed set can be written as the intersection of a family of
zero set In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or e ...
s 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 set In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or eq ...
s of $X$ form a
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
for the topology of $X.$ Given an arbitrary topological space $\left(X, \tau\right)$ there is a universal way of associating a completely regular space with $\left(X, \tau\right).$ Let ρ be the initial topology on $X$ induced by $C_\left(X\right)$ or, equivalently, the topology generated by the basis of cozero sets in $\left(X, \tau\right).$ Then ρ will be the finest completely regular topology on $X$ that is coarser than $\tau.$ This construction is
universal Universal is the adjective for universe. Universal may also refer to: Companies * NBCUniversal, a media and entertainment company ** Universal Animation Studios, an American Animation studio, and a subsidiary of NBCUniversal ** Universal TV, a ...
in the sense that any continuous function $f : (X, \tau) \to Y$ to a completely regular space $Y$ will be continuous on $\left(X, \rho\right).$ In the language of category theory, the
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
that sends $\left(X, \tau\right)$ to $\left(X, \rho\right)$ is
left adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
to the inclusion functor CReg → Top. Thus the category of completely regular spaces CReg is a
reflective subcategory In mathematics, a full subcategory ''A'' of a category ''B'' is said to be reflective in ''B'' when the inclusion functor from ''A'' to ''B'' has a left adjoint. This adjoint is sometimes called a ''reflector'', or ''localization''. Dually, ''A ...
of Top, the category of topological spaces. By taking
Kolmogorov quotient In topology and related branches of mathematics, a topological space ''X'' is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of ''X'', at least one of them has a neighborhood not containing t ...
s, one sees that the subcategory of Tychonoff spaces is also reflective. One can show that $C_\left(X\right) = C_\left(X\right)$ in the above construction so that the rings $C\left(X\right)$ and $C_b\left(X\right)$ are typically only studied for completely regular spaces $X.$ The category of
realcompact In mathematics, in the field of topology, a topological space is said to be realcompact if it is completely regular Hausdorff and every point of its Stone–Čech compactification is real (meaning that the quotient field at that point of the ring ...
Tychonoff spaces is anti-equivalent to the category of the rings $C\left(X\right)$ (where $X$ is realcompact) together with ring homomorphisms as maps. For example one can reconstruct $X$ from $C\left(X\right)$ 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 In mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them (in particular real polynomia ...
, is the class of real closed rings.

## Embeddings

Tychonoff spaces are precisely those spaces that can be embedded in
compact Hausdorff space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i. ...
s. 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 In mathematics, more specifically in general topology, the Tychonoff cube is the generalization of the unit cube from the product of a finite number of unit intervals to the product of an infinite, even uncountable number of unit intervals. The Tyc ...
(i.e. a possibly infinite product of
unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis, ...
s). Every Tychonoff cube is compact Hausdorff as a consequence of
Tychonoff's theorem In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is trans ...
. 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 Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
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 In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a universal map from a topological space ''X'' to a compact Hausdorff space ''βX''. The Ston ...
$\beta X.$ It is characterized by the
universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
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 Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
of $g$ and $j.$

## Uniform structures

Complete regularity is exactly the condition necessary for the existence of
uniform structure In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifor ...
s on a topological space. In other words, every
uniform space In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifo ...
has a completely regular topology and every completely regular space $X$ is
uniformizable In mathematics, a topological space ''X'' is uniformizable if there exists a uniform structure on ''X'' that induces the topology of ''X''. Equivalently, ''X'' is uniformizable if and only if it is homeomorphic to a uniform space (equipped with ...
. 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 In mathematics, a topological space ''X'' is uniformizable if there exists a uniform structure on ''X'' that induces the topology of ''X''. Equivalently, ''X'' is uniformizable if and only if it is homeomorphic to a uniform space (equipped with ...
on $X.$ If $X$ is Tychonoff, then the uniform structure can be chosen so that $\beta X$ becomes the completion of the uniform space $X.$