Separation axioms in topological spaces | |
---|---|

Kolmogorov classification | |

T_{0} | (Kolmogorov) |

T_{1} | (Fréchet) |

T_{2} | (Hausdorff) |

T_{2½} | (Urysohn) |

completely T_{2} | (completely Hausdorff) |

T_{3} | (regular Hausdorff) |

T_{3½} | (Tychonoff) |

T_{4} | (normal Hausdorff) |

T_{5} | (completely normal Hausdorff) |

T_{6} | (perfectly normal Hausdorff) |

In topology and related branches of mathematics, a **normal space** is a topological space *X* that satisfies **Axiom T _{4}**: every two disjoint closed sets of

A topological space *X* is a **normal space** if, given any disjoint closed sets *E* and *F*, there are neighbourhoods *U* of *E* and *V* of *F* that are also disjoint. More intuitively, this condition says that *E* and *F* can be separated by neighbourhoods.

A **T _{4} space** is a T

A **completely normal space** or a **hereditarily normal space** is a topological space *X* such that every subspace of *X* with subspace topology is a normal space. It turns out that *X* is completely normal if and only if every two separated sets can be separated by neighbourhoods. Also, *X* is completely normal if and only if every open subset of *X* is normal with the subspace topology.

A **completely T _{4} space**, or

A **perfectly normal space** is a topological space *X* in which every two disjoint closed sets *E* and *F* can be precisely separated by a continuous function *f* from *X* to the real line **R**: the preimages of {0} and {1} under *f* are, respectively, *E* and *F*. (In this definition, the real line can be replaced with the unit interval [0,1].)

It turns out that *X* is perfectly normal if and only if *X* is normal and every closed set is a G_{δ} set. Equivalently, *X* is perfectly normal if and only if every closed set is a zero set. Every perfectly normal space is automatically completely normal.^{ [1] }

A Hausdorff perfectly normal space *X* is a **T _{6} space**, or

Note that the terms "normal space" and "T_{4}" and derived concepts occasionally have a different meaning. (Nonetheless, "T_{5}" always means the same as "completely T_{4}", whatever that may be.) The definitions given here are the ones usually used today. For more on this issue, see History of the separation axioms.

Terms like "normal regular space" and "normal Hausdorff space" also turn up in the literature—they simply mean that the space both is normal and satisfies the other condition mentioned. In particular, a normal Hausdorff space is the same thing as a T_{4} space. Given the historical confusion of the meaning of the terms, verbal descriptions when applicable are helpful, that is, "normal Hausdorff" instead of "T_{4}", or "completely normal Hausdorff" instead of "T_{5}".

Fully normal spaces and fully T_{4} spaces are discussed elsewhere; they are related to paracompactness.

A locally normal space is a topological space where every point has an open neighbourhood that is normal. Every normal space is locally normal, but the converse is not true. A classical example of a completely regular locally normal space that is not normal is the Nemytskii plane.

Most spaces encountered in mathematical analysis are normal Hausdorff spaces, or at least normal regular spaces:

- All metric spaces (and hence all metrizable spaces) are perfectly normal Hausdorff;
- All pseudometric spaces (and hence all pseudometrisable spaces) are perfectly normal regular, although not in general Hausdorff;
- All compact Hausdorff spaces are normal;
- In particular, the Stone–Čech compactification of a Tychonoff space is normal Hausdorff;
- Generalizing the above examples, all paracompact Hausdorff spaces are normal, and all paracompact regular spaces are normal;
- All paracompact topological manifolds are perfectly normal Hausdorff. However, there exist non-paracompact manifolds that are not even normal.
- All order topologies on totally ordered sets are hereditarily normal and Hausdorff.
- Every regular second-countable space is completely normal, and every regular Lindelöf space is normal.

Also, all fully normal spaces are normal (even if not regular). Sierpinski space is an example of a normal space that is not regular.

An important example of a non-normal topology is given by the Zariski topology on an algebraic variety or on the spectrum of a ring, which is used in algebraic geometry.

A non-normal space of some relevance to analysis is the topological vector space of all functions from the real line **R** to itself, with the topology of pointwise convergence. More generally, a theorem of Arthur Harold Stone states that the product of uncountably many non-compact metric spaces is never normal.

Every closed subset of a normal space is normal. The continuous and closed image of a normal space is normal.^{ [2] }

The main significance of normal spaces lies in the fact that they admit "enough" continuous real-valued functions, as expressed by the following theorems valid for any normal space *X*.

Urysohn's lemma: If *A* and *B* are two disjoint closed subsets of *X*, then there exists a continuous function *f* from *X* to the real line **R** such that *f*(*x*) = 0 for all *x* in *A* and *f*(*x*) = 1 for all *x* in *B*. In fact, we can take the values of *f* to be entirely within the unit interval [0,1]. (In fancier terms, disjoint closed sets are not only separated by neighbourhoods, but also separated by a function.)

More generally, the Tietze extension theorem: If *A* is a closed subset of *X* and *f* is a continuous function from *A* to **R**, then there exists a continuous function *F*: *X* → **R** that extends *f* in the sense that *F*(*x*) = *f*(*x*) for all *x* in *A*.

If **U** is a locally finite open cover of a normal space *X*, then there is a partition of unity precisely subordinate to **U**. (This shows the relationship of normal spaces to paracompactness.)

In fact, any space that satisfies any one of these three conditions must be normal.

A product of normal spaces is not necessarily normal. This fact was first proved by Robert Sorgenfrey. An example of this phenomenon is the Sorgenfrey plane. In fact, since there exist spaces which are Dowker, a product of a normal space and [0, 1] need not to be normal. Also, a subset of a normal space need not be normal (i.e. not every normal Hausdorff space is a completely normal Hausdorff space), since every Tychonoff space is a subset of its Stone–Čech compactification (which is normal Hausdorff). A more explicit example is the Tychonoff plank. The only large class of product spaces of normal spaces known to be normal are the products of compact Hausdorff spaces, since both compactness (Tychonoff's theorem) and the T_{2} axiom are preserved under arbitrary products.^{ [3] }

If a normal space is R_{0}, then it is in fact completely regular. Thus, anything from "normal R_{0}" to "normal completely regular" is the same as what we usually call *normal regular*. Taking Kolmogorov quotients, we see that all normal T_{1} spaces are Tychonoff. These are what we usually call *normal Hausdorff* spaces.

A topological space is said to be pseudonormal if given two disjoint closed sets in it, one of which is countable, there are disjoint open sets containing them. Every normal space is pseudonormal, but not vice versa.

Counterexamples to some variations on these statements can be found in the lists above. Specifically, Sierpinski space is normal but not regular, while the space of functions from **R** to itself is Tychonoff but not normal.

- ↑ Munkres 2000 , p. 213
- ↑ Willard 1970, pp. 100–101.
- ↑ Willard 1970, Section 17.

In topology and related branches of mathematics, a **Hausdorff space**, **separated space** or **T _{2} space** is a topological space where for any two distinct points there exist neighbourhoods of each which are disjoint from each other. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T

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 obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; in that sense the product topology is the natural topology on the Cartesian product.

In mathematics, a topological space is called **separable** if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

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.

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

In mathematics, a **paracompact space** is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by Dieudonné (1944). Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it admits partitions of unity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff.

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, **Urysohn's lemma** is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a continuous function.

In the mathematical discipline of general topology, **Stone–Čech compactification** is a technique for constructing a universal map from a topological space *X* to a compact Hausdorff space *βX*. The Stone–Čech compactification *βX* of a topological space *X* is the largest, most general compact Hausdorff space "generated" by *X*, in the sense that any continuous map from *X* to a compact Hausdorff space factors through *βX*. If *X* is a Tychonoff space then the map from *X* to its image in *βX* is a homeomorphism, so *X* can be thought of as a (dense) subspace of *βX*; every other compact Hausdorff space that densely contains *X* is a quotient of *βX*. For general topological spaces *X*, the map from *X* to *βX* need not be injective.

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* can be separated by neighborhoods. This condition is known as **Axiom T _{3}**. The term "

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, who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the remark that its proof was the same as for the special case. The earliest known published proof is contained in a 1937 paper of Eduard Čech.

In mathematics, **general topology** is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is **point-set topology**.

In topology, a discipline within mathematics, an **Urysohn space**, or **T _{2½} space**, is a topological space in which any two distinct points can be separated by closed neighborhoods. A

In topology, a topological space with the **trivial topology** is one where the only open sets are the empty set and the entire space. Such spaces are commonly called **indiscrete**, **anti-discrete**, or **codiscrete**. Intuitively, this has the consequence that all points of the space are "lumped together" and cannot be distinguished by topological means. Every indiscrete space is a pseudometric space in which the distance between any two points is zero.

In topology and related areas of mathematics, a **topological property** or **topological invariant** is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space *X* possesses that property every space homeomorphic to *X* possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.

In topology, a branch of mathematics, a **topological manifold** is a topological space which locally resembles real *n*-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold. Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure.

In mathematics, the **Moore plane**, also sometimes called **Niemytzki plane**, is a topological space. It is a completely regular Hausdorff space that is not normal. It is named after Robert Lee Moore and Viktor Vladimirovich Nemytskii.

In mathematics, particularly topology, a **G _{δ} space** is a topological space in which closed sets are in a way ‘separated’ from their complements using only countably many open sets. A G

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 called *Tychonoff separation axioms*, after Andrey Tychonoff.

- Kemoto, Nobuyuki (2004). "Higher Separation Axioms". In K.P. Hart; J. Nagata; J.E. Vaughan (eds.).
*Encyclopedia of General Topology*. Amsterdam: Elsevier Science. ISBN 978-0-444-50355-8. - Munkres, James R. (2000).
*Topology*(2nd ed.). Prentice-Hall. ISBN 978-0-13-181629-9. - Sorgenfrey, R.H. (1947). "On the topological product of paracompact spaces".
*Bull. Amer. Math. Soc*.**53**(6): 631–632. doi: 10.1090/S0002-9904-1947-08858-3 . - Stone, A. H. (1948). "Paracompactness and product spaces".
*Bull. Amer. Math. Soc*.**54**(10): 977–982. doi: 10.1090/S0002-9904-1948-09118-2 . - Willard, Stephen (1970).
*General Topology*. Reading, MA: Addison-Wesley. ISBN 978-0-486-43479-7.

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