Home > Urysohn's Lemma

if X is a normal topological space and A and B are disjoint closed subsets of X, then there exists a continuous function from X into the unit interval [0, 1],

f : X → [0, 1],

such that f(a) = 0 for all a in A and f(b) = 1 for all b in B.

The lemma, sometimes called "the first non-trivial fact of point set topology", is often used to construct continuous functions with various properties; 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.

Note that in the statement above, we do not, and in general cannot, require that f(x) ≠ 0 and ≠ 1 for x outside of A and B. This is only possible in perfectly normal spaces.

The lemma is named after Paul Samuilovich Urysohn,

Proof sketch

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

1. U(r) contains A and is disjoint from B for all r
2. for r < s, the closureIn topology and mathematical analysis, the closure of a subset of a topological space is the smallest closed subset of which contains. This can be constructed by intersecting all closed supersets of in. Notation The closure of is written as or. If there i of U(r) is contained in U(s)

Once we have these sets, we define f(x) = infIn mathematics the infimum of a subset of some set is the greatest element that is smaller than all other elements of the subset. Consequently the term greatest lower bound is also commonly used. Infima of real numbers are a common special case that is es { r : x ∈ U(r) } for every x ∈ X. Using the fact that the dyadic rationals are denseTopology General topology In mathematics, the term dense has at least two different meanings. A subset A of a topological space X is said to be dense if the only closed subset of X containing A is X itself. This can also be expressed by saying that the cl, 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 inductionMathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers, or otherwise is true of all members of an infinite sequence. A somewhat more general form of argument used in mathe. 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(a/2ⁿ) and V(a/2ⁿ) have already been constructed for a = 1,...,2ⁿ-1. Since X is normal, we can find two disjoint open sets which contain the complement of V(a/2ⁿ) and the complement of U((a+1)/2ⁿ), respectively. Call these two open sets U((2a+1)/2ⁿ⁺¹) and V((2a+1)/2ⁿ⁺¹), and verify the above three conditions.

The Mizar projectThe Mizar system consists of a language for writing strictly formalized mathematical definitions and proofs, a computer program which is able to check proofs written in this language, and a library of definitions and proved theorems which can be referred has completely formalized and automatically checked a proof of Urysohn's lemma in the URYSOHN3 file.