View source for Absolute neighbourhood extensor

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For the time being, assume that all topological spaces under discussion are metrizable (cf. also [[Metrizable space|Metrizable space]]). A space $X$ is called an absolute (neighbourhood) extensor, abbreviated AE (respectively, ANE), provided that for every space $Y$ and every closed subspace $A \subset Y$, every [[Continuous function|continuous function]] $f : A \rightarrow X$ can be extended over $Y$ (respectively, over a neighbourhood of $A$ in $Y$). The classical Tietze extension theorem (cf. also [[Extension theorems|Extension theorems]]) implies that familiar spaces such as the real line $\mathbf{R}$, the unit interval $\mathbf{I}$ and the circle $S ^ { 1 }$ are absolute (neighbourhood) extensors. An absolute (neighbourhood) retract is a space $X$ having the property that whenever $X$ is embedded as a closed subset of a space $Y$, then it is a (neighbourhood) retract of $Y$ (cf. also [[Absolute retract for normal spaces|Absolute retract for normal spaces]]; [[Retract of a topological space|Retract of a topological space]]). It is a fundamental theorem that every AE (respectively, ANE) is an AR (respectively, ANR), and conversely. The theory of absolute (neighbourhood) retracts was initiated by K. Borsuk in [[#References|[a1]]], [[#References|[a2]]]. He proved his fundamental homotopy extension theorem in [[#References|[a3]]]: If $A$ is a closed subspace of a space $X$ and $Z$ is an ANR and $H : A \times \mathbf{I} \rightarrow Z$ is a homotopy such that $H _ { 0 }$ is extendable to a function $f : X \rightarrow Z$, then there is a homotopy $F : X \times \mathbf{I} \rightarrow Z$ such that $F _ { 0 } = f$, and for every $t \in \mathbf{I}$, $F _ { t } | _ { A } = H _ { t }$. For more details, see [[#References|[a4]]] and [[#References|[a10]]].

In 1951, J. Dugundji [[#References|[a9]]] proved that a [[Convex set|convex set]] in a locally convex vector space (cf. also [[Locally convex space|Locally convex space]]) is an AR. This result was a major improvement over the Tietze extension theorem and was widely applied. The fundamental problem whether the local convexity assumption in this result could be dropped, was solved by R. Cauty [[#References|[a5]]] in the negative. His counterexample used in an essential way a theorem of A.N. Dranishnikov [[#References|[a8]]] about the existence of an infinite-dimensional compactum with finite cohomological dimension.

There are several topological characterizations of absolute (neighbourhood) retracts. The most useful one is due to S. Lefschetz [[#References|[a11]]] and is in terms of partial realizations of polytopes in the space under consideration. It can be shown that if a space $X$ is dominated (cf. also [[Homotopy type|Homotopy type]]) by a [[Simplicial complex|simplicial complex]], then it has the homotopy type of another simplicial complex (see [[#References|[a12]]]). Since it is not too hard to prove that every ANR is dominated by a simplicial complex (see [[#References|[a10]]]), it follows, in particular, that every ANR has the homotopy type of some simplicial complex. But the natural question whether every compact ANR has the homotopy type of a compact simplicial complex, i.e. a finite [[Polyhedron|polyhedron]], remained unanswered for a long time. It was finally solved in the affirmative by J.E. West [[#References|[a17]]] by using powerful results from T.A. Chapman [[#References|[a6]]] in infinite-dimensional topology.

Another fundamental problem about absolute (neighbourhood) retracts was Borsuk's problem of whether for compact absolute (neighbourhood) retracts $X$ and $Y$, the topological dimension of $X \times Y$ is equal to the sum of the dimensions of $X$ and $Y$, respectively. This problem was solved by Dranishnikov [[#References|[a7]]], who proved that there exist $4$-dimensional compact absolute retracts $X$ and $Y$ whose product is of dimension $7$.

The theory of absolute (neighbourhood) retracts played a key role in infinite-dimensional topology. The fundamental topological characterization results of manifolds over the Hilbert cube and the Hilbert space, respectively, which are due to H. Toruńczyk [[#References|[a15]]], [[#References|[a16]]], are stated in terms of absolute (neighbourhood) retracts. Many of the remaining open problems in infinite-dimensional topology have been proven to actually be problems about absolute (neighbourhood) retracts.

The theory of absolute (neighbourhood) retracts in Tikhonov spaces (cf. also [[Tikhonov space|Tikhonov space]]). was mainly considered by E.V. Shchepin. He proved in [[#References|[a13]]] that finite-dimensional compact absolute (neighbourhood) retracts are metrizable. He also found, in [[#References|[a14]]], a very interesting topological characterization of all Tikhonov cubes of uncountable weight. ANR-theory plays a crucial role in this characterization.

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  K. Borsuk,   "Sur les rétractes"  ''Fund. Math.'' , '''17'''  (1931)  pp. 152–170</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  K. Borsuk,   "Über eine Klasse von lokal zusammenhängenden Räumen"  ''Fund. Math.'' , '''19'''  (1932)  pp. 220–242</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  K. Borsuk,   "Sur les prolongements des transformations continus"  ''Fund. Math.'' , '''28'''  (1936)  pp. 99–110</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  K. Borsuk,   "Theory of retracts" , PWN  (1967)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  R. Cauty,   "Un espace métrique linéaire qui n'est pas un rétracte absolu"  ''Fund. Math.'' , '''146'''  (1994)  pp. 85–99</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  T.A. Chapman,   "Lectures on Hilbert cube manifolds" , ''CBMS'' , '''28''' , Amer. Math. Soc.  (1975)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  A.N. Dranišnikov,   "On the dimension of the product of ANR-compacta"  ''Dokl. Akad. Nauk SSSR'' , '''300''' :  5  (1988)  pp. 1045–1049</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  A.N. Dranišnikov,   "On a problem of P.S. Alexandrov"  ''Mat. Sb.'' , '''135'''  (1988)  pp. 551–557</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  J. Dugundji,   "An extension of Tietze's theorem"  ''Pac. J. Math.'' , '''1'''  (1951)  pp. 353–367</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  S.T. Hu,   "Theory of retracts" , Wayne State Univ. Press  (1965)</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  S. Lefschetz,   "On compact spaces"  ''Ann. of Math.'' , '''32'''  (1931)  pp. 521–538</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  A.T. Lundell,   S. Weingram,   "The topology of CW-complexes" , Litton  (1969)</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  E.V. Shchepin,   "Finite-dimensional bicompact absolute neighborhood retracts are metrizable"  ''Dokl. Akad. Nauk SSSR'' , '''233'''  (1977)  pp. 304–307  (In Russian)</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  E.V. Shchepin,   "On Tychonoff manifolds"  ''Dokl. Akad. Nauk SSSR'' , '''246'''  (1979)  pp. 551–554  (In Russian)</td></tr><tr><td valign="top">[a15]</td> <td valign="top">  H. Toruńczyk,   "On $C E$-images of the Hilbert cube and characterizations of $Q$-manifolds"  ''Fund. Math.'' , '''106'''  (1980)  pp. 31–40</td></tr><tr><td valign="top">[a16]</td> <td valign="top">  H. Toruńczyk,   "Characterizing Hilbert space topology"  ''Fund. Math.'' , '''111'''  (1981)  pp. 247–262</td></tr><tr><td valign="top">[a17]</td> <td valign="top">  J.E. West,   "Mapping Hilbert cube manifolds to ANR's: a solution to a conjecture of Borsuk"  ''Ann. of Math.'' , '''106'''  (1977)  pp. 1–18</td></tr></table>