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
and
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 ...
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
of
and
of
that are disjoint. In particular
and
are necessarily disjoint.
Two plain subsets
and
are said to be
separated by a continuous function if there exists a
continuous function