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.^{ [1] }

Urysohn's lemma is commonly used to construct continuous functions with various properties on normal spaces. It is widely applicable since all metric spaces and all compact Hausdorff spaces are normal. The lemma is generalized by (and usually used in the proof of) the Tietze extension theorem.

The lemma is named after the mathematician Pavel Samuilovich Urysohn.

Two subsets *A* and *B* of a topological space *X* are said to be separated by neighbourhoods if there are neighbourhoods *U* of *A* and *V* of *B* that are disjoint. In particular *A* and *B* are necessarily disjoint.

Two plain subsets *A* and *B* are said to be separated by a function if there exists a continuous function *f* from *X* into the unit interval [0,1] such that *f*(*a*) = 0 for all *a* in *A* and *f*(*b*) = 1 for all *b* in *B*. Any such function is called a **Urysohn function** for *A* and *B*. In particular *A* and *B* are necessarily disjoint.

It follows that if two subsets *A* and *B* are separated by a function then so are their closures.

Also it follows that if two subsets *A* and *B* are separated by a function then *A* and *B* are separated by neighbourhoods.

A normal space is a topological space in which any two disjoint closed sets can be separated by neighbourhoods. Urysohn's lemma states that a topological space is normal if and only if any two disjoint closed sets can be separated by a continuous function.

The sets *A* and *B* need not be precisely separated by *f*, i.e., we do not, and in general cannot, require that *f*(*x*) ≠ 0 and ≠ 1 for *x* outside of *A* and *B*. The spaces in which this property holds are the perfectly normal spaces.

Urysohn's lemma has led to the formulation of other topological properties such as the 'Tychonoff property' and 'completely Hausdorff spaces'. For example, a corollary of the lemma is that normal *T*_{1} spaces are Tychonoff.

A topological space *X* is normal if and only if, for any two non-empty closed disjoint subsets *A* and *B* of *X*, there exists a continuous map such that and .

The procedure is an entirely straightforward application of the definition of normality (once one draws some figures representing the first few steps in the induction described below to see what is going on), beginning with two disjoint closed sets. The *clever* part of the proof is the indexing of the open sets thus constructed by dyadic fractions.

For every dyadic fraction *r* ∈ (0,1), we are going to construct an open subset *U*(*r*) of *X* such that:

*U*(*r*) contains*A*and is disjoint from*B*for all*r*,- For
*r*<*s*, the closure of*U*(*r*) is contained in*U*(*s*).

Once we have these sets, we define *f*(*x*) = 1 if *x* ∉ *U*(*r*) for any *r*; otherwise *f*(*x*) = inf{ *r* : *x* ∈ *U*(*r*) } for every *x* ∈ *X*. Using the fact that the dyadic rationals are dense, it is then not too hard to show that *f* is continuous and has the property *f*(*A*) ⊆ {0} and *f*(*B*) ⊆ {1}.

In order to construct the sets *U*(*r*), we actually do a little bit more: we construct sets *U*(*r*) and *V*(*r*) such that

*A*⊆*U*(*r*) and*B*⊆*V(r)*for all*r*,*U*(*r*) and*V*(*r*) are open and disjoint for all*r*,- For
*r*<*s*,*V*(*s*) is contained in the complement of*U*(*r*) and the complement of*V*(*r*) is contained in*U*(*s*).

Since the complement of *V*(*r*) is closed and contains *U*(*r*), the latter condition then implies condition (2) from above.

This construction proceeds by mathematical induction. First define *U*(1) = *X* \ *B* and *V*(0) = *X* \ *A*. Since *X* is normal, we can find two disjoint open sets *U*(1/2) and *V*(1/2) which contain *A* and *B*, respectively. Now assume that *n* ≥ 1 and the sets *U*(*k* / 2^{n}) and *V*(*k* / 2^{n}) have already been constructed for *k* = 1, ..., 2^{n}−1. Since *X* is normal, for any *a* ∈ { 0, 1, ..., 2^{n}−1 }, we can find two disjoint open sets which contain *X* \ *V*(*a* / 2^{n}) and *X* \ *U*((*a*+1) / 2^{n}), respectively. Call these two open sets *U*((2*a*+1) / 2^{n+1}) and *V*((2*a*+1) / 2^{n+1}), and verify the above three conditions.

The Mizar project has completely formalized and automatically checked a proof of Urysohn's lemma in the URYSOHN3 file.

- ↑ Willard 1970 Section 15.

In mathematics, more specifically in general topology, **compactness** is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space in various ways.

In topology and related branches of mathematics, a **connected space** is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces.

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

The **Hahn–Banach theorem** is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting". Another version of the Hahn–Banach theorem is known as the **Hahn–Banach separation theorem** or the hyperplane separation theorem, and has numerous uses in convex geometry.

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.

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 topology, the **Tietze extension theorem** states that continuous functions on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary.

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

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, **separated sets** are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets are separated or not is important both to the notion of connected spaces as well as to the separation axioms for topological spaces.

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 and related areas of mathematics, the **quotient space** of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the **quotient topology**, that is, with the finest topology that makes continuous the canonical projection map. In other words, a subset of a quotient space is open if and only if its preimage under the canonical projection map is open in the original topological space.

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 mathematics, the **Sierpiński space** is a finite topological space with two points, only one of which is closed. It is the smallest example of a topological space which is neither trivial nor discrete. It is named after Wacław Sierpiński.

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 the mathematical field of topology, the **inductive dimension** of a topological space *X* is either of two values, the **small inductive dimension** ind(*X*) or the **large inductive dimension** Ind(*X*). These are based on the observation that, in *n*-dimensional Euclidean space *R*^{n}, (*n* − 1)-dimensional spheres have dimension *n* − 1. Therefore it should be possible to define the dimension of a space inductively in terms of the dimensions of the boundaries of suitable open sets.

In topology and other branches of mathematics, a topological space *X* is **locally connected** if every point admits a neighbourhood basis consisting entirely of open, connected sets.

In topology and related areas of mathematics, a subset *A* of a topological space *X* is called **dense** if every point *x* in *X* either belongs to *A* or is a limit point of *A*; that is, the closure of *A* constitutes the whole set *X*. Informally, for every point in *X*, the point is either in *A* or arbitrarily "close" to a member of *A* — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it.

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.

In mathematics, a **polyadic space** is a topological space that is the image under a continuous function of a topological power of an Alexandroff one-point compactification of a discrete space.

- Willard, Stephen (1970).
*General Topology*. Dover Publications. ISBN 0-486-43479-6.

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