The Katětov–Tong insertion theorem is a theorem of

point-set topology In mathematics, general topology (or point set 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 differ ...

proved independently by

Miroslav Katětov Miroslav Katětov (; March 17, 1918, Chembar, Russia – December 15, 1995) was a Czech mathematician, chess master, and psychologist. His research interests in mathematics included topology and functional analysis. He was an author of the Katě ...

and

Hing Tong Hing Tong (16 February 1922 – 4 March 2007) was an American mathematician. He is well known for providing the original proof of the Katetov–Tong insertion theorem. Life Hing Tong was born in Guangzhou, Canton, China. He received his bachelor' ...

in the 1950s. The theorem states the following: Let

X

be a

normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...

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 ...

and let

g, h\colon X \to \mathbb

be functions with

g

upper

semicontinuous In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semicontinuous at a point x_0 if, ro ...

h

lower semicontinuous, and

g \leq h

. Then there exists a continuous function

f\colon X \to \mathbb

with

g \leq f \leq h.

This theorem has a number of applications and is the first of many classical insertion theorems. In particular it implies 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 ...

and consequently

Urysohn's lemma 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. Section 15. Urysohn's lemma is commonly used to construct contin ...

, and so the conclusion of the theorem is equivalent to normality.

References

* {{DEFAULTSORT:Katetov-Tong insertion theorem General topology Theorems in topology