Tietze extension theorem explained
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 can be extended to the entire space, preserving boundedness if necessary.
Formal statement
If
is a
normal space and
is a continuous map from a
closed subset
of
into the
real numbers
carrying the
standard topology, then there exists a of
to
that is, there exists a map
continuous on all of
with
for all
Moreover,
may be chosen such that
that is, if
is bounded then
may be chosen to be bounded (with the same bound as
).
Proof
The function
is constructed iteratively. Firstly, we define
Observe that
and
are
closed and
disjoint subsets of
. By taking a linear combination of the function obtained from the proof of
Urysohn's lemma, there exists a
continuous function
such that
and furthermore
on
. In particular, it follows that
on
. We now use
induction to construct a sequence of continuous functions
such that
We've shown that this holds for
and assume that
have been constructed. Define
and repeat the above argument replacing
with
and replacing
with
. Then we find that there exists a continuous function
such that
By the inductive hypothesis,
hence we obtain the required identities and the induction is complete. Now, we define a continuous function
as
Given
,
Therefore, the sequence
is
Cauchy. Since the
space of continuous functions on
together with the
sup norm is a
complete metric space, it follows that there exists a continuous function
such that
converges uniformly to
. Since
on
, it follows that
on
. Finally, we observe that
hence
is bounded and has the same bound as
.
History
L. E. J. Brouwer and Henri Lebesgue proved a special case of the theorem, when
is a finite-dimensional real
vector space.
Heinrich Tietze extended it to all
metric spaces, and
Pavel Urysohn proved the theorem as stated here, for normal topological spaces.
[1] Equivalent statements
This theorem is equivalent to Urysohn's lemma (which is also equivalent to the normality of the space) and is widely applicable, since all metric spaces and all compact Hausdorff spaces are normal. It can be generalized by replacing
with
for some indexing set
any retract of
or any normal absolute retract whatsoever.
Variations
If
is a metric space,
a non-empty subset of
and
is a
Lipschitz continuous function with Lipschitz constant
then
can be extended to a Lipschitz continuous function
with same constant
This theorem is also valid for
Hölder continuous functions, that is, if
is Hölder continuous function with constant less than or equal to
then
can be extended to a Hölder continuous function
with the same constant.
[2] Another variant (in fact, generalization) of Tietze's theorem is due to H.Tong and Z. Ercan:[3] Let
be a closed subset of a normal topological space
If
is an upper semicontinuous function,
a lower semicontinuous function, and
a continuous function such that
for each
and
for each
, then there is a continuousextension
of
such that
for each
This theorem is also valid with some additional hypothesis if
is replaced by a general locally solid
Riesz space.
Dugundji (1951) extends the theorem as follows: If
is a metric space,
is a
locally convex topological vector space,
is a closed subset of
and
is continuous, then it could be extended to a continuous function
defined on all of
. Moreover, the extension could be chosen such that
\tildef(X)\subseteqconvf(A)
External links
Notes and References
- .
- McShane. E. J.. Extension of range of functions. Bulletin of the American Mathematical Society. 1 December 1934. 40. 12. 837–843. 10.1090/S0002-9904-1934-05978-0. free.
- Zafer. Ercan. Extension and Separation of Vector Valued Functions. Turkish Journal of Mathematics. 1997. 21. 4. 423–430.