Urysohn's Lemma
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In
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, Urysohn's lemma is a lemma that states that a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
is normal if and only if any two disjoint closed subsets can be separated by a continuous function. Section 15. Urysohn's lemma is commonly used to construct continuous functions with various properties on normal spaces. It is widely applicable since all
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
s and all compact
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...
s are normal. The lemma is generalised by (and usually used in the proof of) the
Tietze extension theorem In topology, the Tietze extension theorem (also known as the Tietze– Urysohn– Brouwer extension theorem or Urysohn-Brouwer lemma) states that any real-valued, continuous function on a closed subset of a normal topological space In mathe ...
. The lemma is named after the
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
Pavel Samuilovich Urysohn.


Discussion

Two subsets A and B of a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
X are said to be separated by neighbourhoods if there are
neighbourhood A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neighbourh ...
s 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 continuous function if there exists a continuous function f : X \to , 1/math> from X into the unit interval , 1/math> 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 Normal(s) or The Normal(s) may refer to: Film and television * Normal (2003 film), ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * Normal (2007 film), ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keit ...
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., it is not necessary and guaranteed that f(x) \neq 0 and \neq 1 for x outside A and B. A topological space X in which every two disjoint closed subsets A and B are precisely separated by a continuous function is perfectly normal. 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 T1 spaces are Tychonoff.


Formal statement

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 f : X \to , 1/math> such that f(A) = \ and f(B) = \.


Proof sketch

The proof proceeds by repeatedly applying the following alternate characterization of normality. If X is a normal space, Z is an open subset of X, and Y\subseteq Z is closed, then there exists an open U and a closed V such that Y\subseteq U\subseteq V\subseteq Z. Let A and B be disjoint closed subsets of X. The main idea of the proof is to repeatedly apply this characterization of normality to A and B^\complement, continuing with the new sets built on every step. The sets we build are indexed by dyadic fractions. For every dyadic fraction r \in (0, 1), we construct an open subset U(r) and a closed subset V(r) of X such that: * A \subseteq U(r) and V(r)\subseteq B^\complement for all r, * U(r)\subseteq V(r) for all r, * For r < s, V(r)\subseteq U(s). Intuitively, the sets U(r) and V(r) expand outwards in layers from A: : \begin A&&&&&&&\subseteq&&&&&&& B^\complement\\ A&&&\subseteq&&&\ U(1/2)&\subseteq& V(1/2)&&&\subseteq&&& B^\complement\\ A&\subseteq& U(1/4)&\subseteq& V(1/4)&\subseteq& U(1/2)&\subseteq& V(1/2)&\subseteq& U(3/4)&\subseteq& V(3/4)&\subseteq& B^\complement \end This construction proceeds by
mathematical induction Mathematical induction is a method for mathematical proof, proving that a statement P(n) is true for every natural number n, that is, that the infinitely many cases P(0), P(1), P(2), P(3), \dots  all hold. This is done by first proving a ...
. For the base step, we define two extra sets U(1) = B^\complement and V(0) = A . Now assume that n \geq 0 and that the sets U\left(k/2^n\right) and V\left(k/2^n\right) have already been constructed for k \in\. Note that this is vacuously satisfied for n=0. Since X is normal, for any a \in \left\, we can find an open set and a closed set such that : V\left(\frac\right)\subseteq U\left(\frac\right)\subseteq V\left(\frac\right)\subseteq U\left(\frac\right) The above three conditions are then verified. Once we have these sets, we define f(x) = 1 if x \not\in U(r) for any r; otherwise f(x) = \inf \ for every x \in X, where \inf denotes the infimum. 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) \subseteq \ and f(B) \subseteq \. This step requires the V(r) sets in order to work. The Mizar project has completely formalised and automatically checked a proof of Urysohn's lemma in th
URYSOHN3 file


See also

* Mollifier


Notes


References

* *


External links

* * Mizar system proof: https://www.mizar.org/version/current/html/urysohn3.html#T20 {{Topology Articles containing proofs Theory of continuous functions Lemmas Separation axioms Theorems in topology