Absolute environmental retreat

The term Absolute Umgebungsretrakt (Engl. Absolute neighborhood retract , short ANR ) is a concept of topology , one of the branches of mathematics that there a whole and particularly in homotopy theory is important.

definition

A topological space is an absolute surrounding retract if the following applies: ${\displaystyle X}$

For every normal closed subspace there is an open environment and a continuous mapping such that applies to all ; So so that the restriction on the identity is. ${\displaystyle Y\subseteq X}$ ${\displaystyle U\supseteq Y\;(U\subseteq X)}$ ${\displaystyle r\colon U\to Y}$ ${\displaystyle r(y)=y}$ ${\displaystyle y\in Y}$ ${\displaystyle r|_{Y}}$ ${\displaystyle Y}$

The definition can also be summarized as follows:

A topological space is an absolute surrounding retract if and only if: ${\displaystyle X}$

If a normal space is a closed subspace and a continuous mapping, whatever it is, there is always a continuous continuation to an environment . ${\displaystyle Z}$ ${\displaystyle Y\subseteq Z}$ ${\displaystyle g\colon Y\rightarrow X}$ ${\displaystyle Z,Y,g}$ ${\displaystyle g}$ ${\displaystyle g^{+}\colon V\rightarrow X}$ ${\displaystyle V\supseteq Y\;(V\subseteq Z)}$

The concept of the absolute surrounding retract goes back to the Polish mathematician Karol Borsuk . However, it is generalized in contemporary mathematics, namely in the way just described.

Examples

• Everyone is an ANR. ${\displaystyle \mathbb {R} ^{n}\;(n\in \mathbb {N} )}$
• The open subsets of an ANR and all retracts within it are ANRs.
• Every separable manifold is an ANR.
• Every locally finite polyhedron is an ANR.
• Theorem of Hanner : If a topological space is covered by a finite number of open subsets, which are all ANRs, then it is an ANR.

properties

• ANRs are locally contractible .
• A space is an ANR if and only if it is a retract of an open subset of a convex subspace of a normalized space .
• Each ANR is homotopy equivalent to a countable CW complex .

literature

• Borsuk: Sur les rétractes . Find. Math. 17, 2-20 (1931).
• Borsuk: Sur un espace compact localement contractile qui n'est pas un rétracte absolu de voisinage. Find. Math. 35, 175-180 (1948).
• Olof Hanner : Some theorems on absolute neighborhood retracts. Arkiv för Matematik, 1, 389-408 (1951), doi : 10.1007 / BF02591376
• John Milnor : On spaces having the homotopy type of a CW complex. Trans. Amer. Math. Soc. 90: 272-280 (1959).
• Horst Schubert : Topology . 4th edition. BG Teubner Verlag , Stuttgart 1975, ISBN 3-519-12200-6 . MR0423277
• Stephen Willard : General Topology (=  Addison-Wesley Series in Mathematics ). Addison-Wesley , Reading, Massachusetts (et al.) 1970. MR0264581

Web links

• Mardešić: Absolute Neighborhood Retracts and Shape Theory

Individual evidence

1. ↑ Horst Schubert: Topology. 1975, p. 158 ff
2. ↑ ^{a b c d} Schubert, op.cit., P. 159
3. ↑ Stephen Willard: General Topology. 1970, p. 106
4. ↑ Milnor, op. Cit., Pp. 272-273
5. ↑ Hanner, op.cit., P. 394
6. ↑ Milnor, op.cit., P. 272