An interesting proof of the nonexistence continuous bijection between $mathbb{R}^n$ and $mathbb{R}^2$ for $nneq 2$}

An in teresting pro of of the nonexistence con tin uous bijection b et w een R n and R 2 for n 6 = 2 F eresh teh Malek ∗ Hamed Daneshpa jouh Hamid Reza Daneshpa jouh † Johannes Hahn ‡ No vem b er 26, 2024 Abstract In this article it is sho wn t ha t there is no contin uous bijection from R n on to R 2 for n 6 = 2 by an elemen tary metho d. This pro of is based on showing that for any cardinal n um b er β ≤ 2 ℵ 0 , there is a partition of R n ( n ≥ 3) into β arcwise connecte d dense subsets. 1 In tro du ction In 1877 Can tor disco v ered a bijection of R on to R n , for an y n ∈ N . Can tor’s map w as discon tin- uous, but the disco v ery of P eano curv e in 1890, sho we d that there existed contin uous (a lthough not injective ) maps of R on to R n . After that and before 1910, se ve ral mathematicians sho w ed that there didn’t exist a bicon tin uous bijection (homeomorphism) f r om R m on to R n , for the cases m = 2 and m = 3 and n > m . F inally in 19 1 1, Brou w er sho w ed that there didn’t exist a homeomorphism b etw ee n R m and R n for n 6 = m (F or a mo dern treatment see Munkres, James (1984), p.109 [2]). The presen t pap er prov es not existance of continous bijection from R n on to R 2 for n 6 = 2 b y an elemen tary metho d. Marey Ellen R udin sho we d [1] tha t for an y coun table cardinal α > 2, w e can not partition the plane in to α arcwise connected dense subsets. In this pap er we sho w tha t for any cardinal n um b er β ≤ 2 ℵ 0 , there is a partition of R n ( n ≥ 3 ) into β ar cwise c onnected de nse subs ets, and then by using this w e sho w that there is no con tin uous bijection from R n on to R 2 , for n 6 = 2. Lemma 1 . Ther e is a p artition of R + into 2 ℵ 0 dense subsets. pro of. Consider the additive group ( R ,+). The quotien t group R / Q has 2 ℵ 0 elemen ts which are dense subse ts of R . In tersect them with R + .  ∗ F acult y o f S ience, K.N.T o osi Univ ersity of T echnology . E mail: mal ek@kntu. ac.ir . † Department of Ma thematics and Computer Science, Univ er sit y o f T ehra n. Email: h.r.da neshpajo uh@khayam.ut.ac.ir . ‡ F aculty of Mathematics and Natura l Sciences, Univ ersity of Rosto c k. 1 Theorem 1 . Ther e is a p artition of R 3 into 2 ℵ 0 ar cwise c on ne cte d dense subsets. pro of. Let { S i | i ∈ I } b e a partition of R + in to 2 ℵ 0 dense subsets. I is just an index set, so w e ma y sup p ose that I = (0 1 ). Define L i = { ( t, it, 0) | t > 0 } and M = ∪ i ∈ I L i and let A i b e the union of all spheres with cente r at the origin and r a dius from S i , i.e A i = { x ∈ R 3 | k x k ∈ S i } . Let B i = ( A i \ M ) ∪ L i . If S is a sphere c ente red at the origin, then S \ M is a sphere with a small arc remo ved, therefore A i \ M is the union o f some arcwise connected punctured spheres, op en half-line L i pastes these punctured spheres together, so B i is arcwise connected. It is ob vious that { B i | i ∈ I } is a partitio n of R 3 with size 2 ℵ 0 . Since S i is dense in R + , A i and consequen tly B i are dense in R 3 .  Corollary 1 . Ther e is a p artition of R n into 2 ℵ 0 ar cwise c on ne cte d dense subsets for n ≥ 3 . pro of. It is enough to set B ( n ) i = B i × R n − 3 , in w hic h B i is as ab o v e. { B ( n ) i | i ∈ I } is a partition of R n whic h satisfies t he claim.  Note that the union of an y num b er of the sets B ( n ) i is an arcwise connected dense subset of R n , hence Corollary 2 . F or a ny c ar di nal n umb er β ≤ 2 ℵ 0 , ther e is a p artition of R n ( n ≥ 3 ) into β ar cwise c on ne cte d dense subsets. Theorem 2 . F or any c ountable c a r d inal α > 2 we c an not p artition the plane into α ar cwise c onne cte d dense subset. pro of. This stateme nt is pro ved in [1 ]  Lemma 2 . L et X , Y b e metric sp ac es and T : X → Y b e a c o ntinuous map ( a ) If A is dense in X an d T is onto, then T ( A ) is dense in Y . ( b ) If B ⊂ X is ar cwise c onne cte d, then T ( B ) is also ar cwise c onne cte d. Theorem 3 . Ther e is no c ontinuous bije ction fr om R onto R m for m 6 = 1 . pro of. Supp ose the con trary , let g : R → R m b e a contin uous bijectiv e map, w e put B n = [ − n, n ] and so we hav e R m = g ( ∪ ∞ n =1 B n ) = ∪ ∞ n =1 g ( B n ). R m is not first category so at least one of the g ( B n ), forexample g ( B k ) has a nonempt y interior in R m , supp ose B ( x, r ) ⊂ g ( B k ). No w w e consider f as a restriction of g to B k , since B k is compact so f : B k → g ( B k ) is a homeomorphism and then B ( x, r ) is homeomorphic with an in terv al in R and that is a con tradiction, b ecause if w e remo ve 3 p oin ts from B ( x, r ) it remains connected but this is not the case f o r the interv als in R  Theorem 4 . Ther e is no c ontinuous bije ction fr om R n onto R 2 for n 6 = 2 pro of. Supp ose the con trary: (a) If n > 2 then according to corollary 2 and lemma 2 w e can partition R 2 in to 3 arcwise connected dense subs ets and this con tradicts theorem 2. (b)If n = 1 t hen this con tradicts theorem 3.  Ac knowle dgmen ts. The authors a r e grateful to the prof ess or Nicola s Hadjisav v as, for his v aluable a dvice s and commen ts. 2 References [1] M.E.RUDIN, Arcwise conne cted sets in the plane, D uk e Math.jour., 30, NO.3 , (1963), PP . 363-366 . [2] Munkres, James (1984 ) . Elemen ts of Algebraic T op ology . Addis on- W esley , Reading, MA. 3