A short and elementary proof of Hanners theorem

A SHOR T AND ELEMENT AR Y PR OOF OF HANNER ’S THEOREM AASA FERAGEN Abstract. Hanner’s theorem is a class ical theorem in the theory o f retracts and extensors in top ological spaces, which states that a lo cal ANE is a n ANE. While Hanner’s origina l pro of of the theorem is quite s imple for separable spaces, it is rather inv olv ed fo r the general case. W e provide a pro of whic h is not only shor t, but also elementary , relying only o n well-known classica l po in t-set top ology . 1. Introduction Denote b y M the class of metrizable spaces. Hanner’s theorem is a fundamen tal theorem in the theory of extensors a nd retracts, stating that a space whic h admits an op en co v ering b y ANEs for M , is an ANE for M . Pro ving that a space with a coun table co v ering b y op en ANEs is an ANE is not hard [2], but the o riginal pro of of the Hanner’s general theorem is r ather complicated [3]. W e giv e a short and elemen tary pro of, ba sed on reducing the uncountable co v ering by ANEs to a coun table co v ering b y ANEs using a tec hnique originating with J. Milnor [4]. Another short pro of of the theorem has b een g iv en b y J. Dydak [1 ] as part of his framew ork for the extension dimension theory . 2. P reliminaries A metrizable space Y is said to b e an ANE for M if, giv en any space X ∈ M and an y con tin uous map f : A → Y where A is a closed subset of X , there exists a neigh b o rho o d U o f A in X a nd a con tin uous extension F : U → Y of f . Theorem 1. i) Any op en s ubset of a sp ac e w hich is an ANE for M is an ANE for M . ii) If X = S i ∈ I U i wher e the U i ar e disjo int op en subsets of X which ar e ANEs for M , then X is an ANE for M . iii) If X = S n ∈ N U n wher e the U n ar e op e n subsets of X which ar e ANEs fo r M , then X is an ANE for M . Pr o of. Claim i) is trivial, and pro o fs of claims ii) , and iii) can b e found in Hanner’s article [2].  3. Hanne r ’s ge neral theorem Our theorem is the follo wing Theorem 2. I f X ∈ M and X = S i ∈ I U i wher e the U i ar e op e n subsets of X which ar e ANEs for M , then X is an ANE for M . Pr o of. Find a par t it ion of unity { ϕ i : X → [0 , 1] } i ∈ I whic h is sub ordinate to the cov ering { U i } i ∈ I of X . F or each finite subset T ⊂ I w e denote W ( T ) = { x ∈ X | ϕ i ( x ) > ϕ j ( x ) ∀ i ∈ T , ∀ j ∈ I \ T } . 2000 Mathematics Subje ct Classific ation. 54C55 . Key wor ds and phr ases. Hanner ’s theorem, ANE, lo c a l ANE.. 1 2 AASA FERAGEN This set is op en b ecause W ( T ) = u − 1 T (0 , 1] for the con tin uo us map u T : X → [0 , 1] , u T ( x ) = max { 0 , min { ϕ i ( x ) − ϕ j ( x ) | i ∈ T , j ∈ I \ T }} . F urthermore, W ( T ) ⊂ ϕ − 1 i (0 , 1] ⊂ U i for eac h i ∈ T since x ∈ W ( T ) implies ϕ i ( x ) > ϕ j ( x ) ≥ 0 f o r eac h i ∈ T and j ∈ I \ T . It follow s tha t W ( T ) is a n ANE fo r M by Theorem 1 part i) . Note that if Card( T ) = Card( T ′ ) and T 6 = T ′ , then W ( T ) ∩ W ( T ′ ) = ∅ , since otherwise for some x ∈ W ( T ) ∩ W ( T ′ ), i ∈ T \ T ′ and j ∈ T ′ \ T we ha v e simultaneous ly ϕ i ( x ) < ϕ j ( x ) and ϕ j ( x ) > ϕ i ( x ), whic h is imp ossible. Define W n = [ { W ( T ) | Card( T ) = n } . Then W n is an ANE for M b y Theorem 1 part ii) . But now X = S n ∈ N W n is an ANE for M b y Theorem 1, part iii) .  4. Ack no wledgements The author wishes to thank the Mag nus Ehrnro oth F oundatio n for financial supp ort, and t he Departmen t of Mathematical Sciences at the Univ ersit y of Aarhus for its hospitality . Reference s [1] J. Dydak, E xtension dimension for par acompact spaces, T op. Appl. 14 0 (2004) 227-2 4 3. [2] O. Hanner , Some theorems on absolute neig hbo rho o d retra c ts, Arkiv f¨ or Matematik , V ol. 1, No. 5 , (1950) 389- 408. [3] S.T. Hu, The ory of R etr acts , 1s t ed. (W ayne State Universit y Press, 1965). [4] R. S. Palais, The classificatio n of G -spaces, Mem. Amer. Math. So c. No. 36 , 1 960. University of Copenhagen, Dep ar tment of Computer Science (DI KU), Universitetsp arken 1, K-2100 Copenhagen , Denmark E-mail addr ess : a asa@di ku.dk