Dugundji extension theorem

In mathematics, the Dugundji extension theorem is a theorem in general topology due to American mathematician James Dugundji. It is directly related to the Tietze–Urysohn extension theorem — about extending continuous functions on normal spaces — of which it is, in a sense, a generalization.

Statement

Let $X$ be a metrizable space, $A$ a closed subset of X, and $L$ a locally convex topological vector space. Then:

• Every continuous map $f:A\to L$ admits a continuous extension $F:X\to L$ such that the image $F(X)$ is contained in the convex hull of $f(A).$

or, equivalently:

• Every continuous map from $A$ into a convex subset $K$ of $L$ admits a continuous extension from $X$ into $K.$

Comparison with the Tietze–Urysohn extension theorem

The first version of the Tietze extension theorem corresponds to the special case of the above theorem where the target space L is the real line ℝ. Urysohn generalized this to replacing the domain being a metric space by an arbitrary normal space. The Dugundji extension theorem is a transverse generalization, replacing the target ℝ by an arbitrary locally convex space. There is another generalization of the Tietze theorem assuming that the domain X is paracompact and the target L is a Banach space.

Proof

Fix some metric $d$ on $X.$ Consider the open cover of $X-A$ that consists of the open balls $B(x,d(x,A)/2)$ for $x\in X-A.$ Since every metric space is paracompact, there exists a locally finite open cover $\{U_i{\}}_{i\in I}$ of $X-A$ such that each $U_i$ is contained in one of those balls. Choose a partition of unity ${\phi }_i$ subordinate to this cover. For each $i$, pick a point $a_i\in A$ satisfying

$d(x_i,a_i)\le 2d(x_i,A),$

which is possible since for each $ϵ>0$, there is an $a\in A$ with $d(x,a)\le d(x,A)+ϵ$. Define the extension $F$ on $X$ by:

$$F(x)=\{\begin{array}{cc}f(x)& \text{if }x\in A,\\ \sum _i{\phi }_i(x)f(a_i)& \text{else.}\end{array}$$

The map $F$ is clearly continuous on $X-A$. We shall then show it is continuous at each point $a$ in $A$ as well. For each $x$ in $U_i$, we have: $2d(x,x_i)\le d(x_i,A)$ or

$d(x,x_i)+2d(x_i,A)\le 5d(x_i,A)-5d(x,x_i).$

Thus, we have:

$d(x,a_i)\le 5d(x,A)$

and then

$d(a,a_i)\le 6d(a,x).$

Now, let a convex neighborhood C of $F(a)$ be given. Then, since $f$ is continuous, there is some $\delta >0$ such that $f(B(a,\delta )\cap A)\subset C$. Then we have $F(B(a,\delta /6))\subset C$ by the above inequality, completing the proof of the continuity. $◻$

Notes and references

Template:Reflist The article started as a machine (ChatGPT) translation of the corresponding article in French Wikipedia [1].

Original work

• Template:Cite journal

Secondary works

• Czesław Bessaga et Aleksander Pełczyński, Selected Topics in Infinite-Dimensional Topology, Warszawa, 1975, p. 57 et s.
• Karol Borsuk, Theory of Retracts, Warszawa, PWN, 1967, p. 77-78.

See also

• Absolute extensor