Separation axiom
A separation axiom is a condition imposed on a topological space that measures the extent to which distinct points, or points and closed sets, can be distinguished by neighborhoods and continuous functions. Separation axioms form a hierarchy rather than a single definition. The weaker conditions describe whether points have different topological behavior, while the stronger conditions control the separation of closed sets and the construction of real-valued functions.
The notation (T_i) derives from the German term Trennungsaxiom. Numbering is not entirely uniform because the properties called regularity and normality do not, by themselves, always include the (T_1) condition. Consequently, (T_3) generally means regular and (T_1), while (T_4) generally means normal and (T_1).
Point-separation conditions
Let (X) be a topological space.
The space (X) is (T_0), or Kolmogorov, when any two distinct points are topologically distinguishable. Thus, for (x\ne y), an open set contains one of the points without containing the other. Equivalently, distinct points have distinct closures. The (T_0) condition removes the redundancy created by points that belong to exactly the same open sets.
The space is (T_1), or Fréchet, when each of two distinct points has a neighborhood that excludes the other. This condition is equivalent to requiring every singleton subset to be closed. It also implies that every finite subset is closed, although arbitrary unions of singletons need not be closed.
A space is (T_2), more commonly called a Hausdorff space, when distinct points possess disjoint open neighborhoods. Hausdorffness implies (T_1). It ensures, among other consequences, that a convergent net or filter has at most one limit. The image of a compact space under a continuous map into a Hausdorff space is closed, and a compact subset of a Hausdorff space is itself closed.
Several conditions occur between Hausdorffness and regularity. A Urysohn space, often denoted (T_{2\frac12}), requires distinct points to have neighborhoods whose closures are disjoint. A completely Hausdorff space requires that distinct points be separated by a continuous real-valued function. These two conditions strengthen Hausdorffness in different ways and are not interchangeable without additional assumptions.
Separation from closed sets
A space is regular when a point and a closed set not containing that point can be enclosed in disjoint open sets. Equivalently, for every point (x) and every neighborhood (U) of (x), there is a neighborhood (V) of (x) satisfying [ x\in V\subseteq \overline V\subseteq U. ] Under the convention most commonly used in the (T_i) hierarchy, a (T_3) space is both regular and (T_1). Regularity alone does not necessarily imply the (T_1) property.
A space is completely regular when points can be separated from closed sets by continuous functions. More precisely, if (F) is closed and (x\notin F), there exists a continuous function [ f:X\longrightarrow [0,1] ] for which (f(x)=0) and (f(y)=1) for every (y\in F). A completely regular (T_1) space is called a Tychonoff space, and the corresponding separation level is commonly denoted (T_{3\frac12}).
Complete regularity connects the topology of (X) with its algebra of continuous real-valued functions. Every Tychonoff space embeds in a product of copies of the unit interval, with the embedding determined by sufficiently many such functions. This characterization places Tychonoff spaces at the natural level of generality for compactification theory.
Separation of closed sets
A space is normal when any two disjoint closed subsets possess disjoint open neighborhoods. The designation (T_4) normally refers to a normal (T_1) space. Normality is a global condition because it compares arbitrary closed subsets, whereas regularity compares one closed subset with a point outside it.
In a normal space, the Urysohn lemma associates continuous functions with pairs of disjoint closed sets. If (A) and (B) are disjoint and closed, there is a continuous map (f:X\to[0,1]) satisfying (f(A)={0}) and (f(B)={1}). It follows that every normal (T_1) space is completely regular.
A space is completely normal when every pair of separated subsets has disjoint neighborhoods. This is equivalent to requiring every subspace to be normal. With the (T_1) condition included, the property is often denoted (T_5).
A perfectly normal space is a normal space in which every closed subset is a (G_\delta) set. Equivalently, every closed subset is the zero set of a continuous real-valued function. Perfect normality is stronger than complete normality under the standard (T_1) conventions, and it is sometimes assigned the symbol (T_6).
Logical structure
With the usual convention that the numbered axioms include the necessary (T_1) condition, the principal implications are [ T_4\Longrightarrow T_{3\frac12}\Longrightarrow T_3 \Longrightarrow T_2\Longrightarrow T_1\Longrightarrow T_0. ] Perfect normality and complete normality impose still stronger requirements than ordinary normality. None of the displayed implications reverses in general.
The hierarchy is not linear once intermediate conditions are included. Functional separation and separation by closures can diverge in spaces lacking regularity assumptions. Definitions that omit (T_1) also produce additional branches because normal or regular spaces may then contain topologically indistinguishable points.
Separation properties are preserved unevenly by standard constructions. Every subspace of a Hausdorff space is Hausdorff, while a subspace of a normal space need not be normal. Arbitrary products of Hausdorff spaces remain Hausdorff, and arbitrary products of Tychonoff spaces remain Tychonoff. A product of normal spaces, however, need not be normal. Quotient spaces may lose even Hausdorffness when the equivalence relation does not satisfy an appropriate closedness condition.
Representative examples
An indiscrete space with more than one point fails (T_0) because its points belong to precisely the same open sets. The Sierpiński space is (T_0) but not (T_1); its two points are distinguishable in only one direction.
An infinite set equipped with the cofinite topology is (T_1), since every singleton is closed. It is not Hausdorff because any two nonempty open sets intersect. This example separates the closedness of points from the existence of disjoint neighborhoods.
The line with two origins is locally modeled on the real line and satisfies (T_1), but its two origins cannot be enclosed in disjoint neighborhoods. It therefore fails the Hausdorff condition despite retaining much of the local structure of a one-dimensional manifold.
The Sorgenfrey line is perfectly normal, whereas its square, the Sorgenfrey plane, is not normal. This supplies a standard failure of product preservation for normality. By contrast, Tychonoff spaces remain Tychonoff under arbitrary products, a fact governed by the product topology and the coordinate functions.
Historical development
The separation hierarchy developed from early attempts to isolate which properties of metric spaces were actually needed in general topology. In 1906, Maurice Fréchet formalized abstract convergence structures and introduced conditions closely related to the later (T_1) axiom. The neighborhood formulation adopted by Felix Hausdorff in 1914 made disjoint neighborhood separation central to the definition of topological space then in use.
During the 1920s, regularity and normality acquired their modern roles through work on extension and metrization problems. In 1937, You Watanabe organized the point-separation and closed-set-separation conditions into an implication diagram that distinguished intrinsic implications from those requiring the additional (T_1) hypothesis. Her classification also standardized the use of counterexamples to record the failure of converse implications.
Subsequent terminology retained more than one convention. In one convention, “regular” and “normal” refer only to the corresponding separation statements. In another, each term incorporates (T_1). The notation (T_3) and (T_4) reduces this ambiguity when its convention is stated explicitly.
Functional and extension theorems
Pavel Urysohn established the continuous-function separation result now called the Urysohn lemma. Its construction produces a nested family of open sets indexed by dyadic rational numbers, from which the separating function is obtained by an infimum. The theorem converts normality into a functional property and forms a central step in the Urysohn metrization theorem.
Heinrich Tietze proved that a continuous real-valued function on a closed subset of a normal space extends continuously to the entire space. The Tietze extension theorem is closely related to the Urysohn lemma and provides a functional characterization of normality for (T_1) spaces.
Andrey Tychonoff developed the product and compactification theory associated with completely regular spaces. The resulting class is exactly the class of spaces that can be embedded in compact Hausdorff spaces. This equivalence links separation by continuous functions with the existence of the Stone–Čech compactification.