TWO TO ONE CONTINUOUS IMAGES OF N. 1. Introduction

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1 TWO TO ONE CONTINUOUS IMAGES OF N ALAN DOW AND GETA TECHANIE Abstract. A fuctio is two-to-oe if every poit i the image has exactly two iverse poits. We show that every two-to-oe cotiuous image of N is homeomorphic to N whe the cotiuum hypothesis is assumed. We also prove that there is o irreducible two-to-oe cotiuous fuctio whose domai is N uder the same assumptio. 1. Itroductio A fuctio f : X Y is two-to-oe if for each y Y, there are exactly two poits of X that map to y. All spaces cosidered are Tychooff. For some spaces X, there does ot exist a two-to-oe cotiuous fuctio f : X Y for ay choice of Y. For example, Harrold [Ha39] showed that there is o two-to-oe cotiuous fuctio f : [0, 1] Y ad Mioduszewski [Mi61] proved that there is o two-to-oe cotiuous fuctio f : R Y. I fact, Heath [He86] later showed that every two-to-oe cotiuous fuctio f : R Y has ifiitely may discotiuities. Aother situatio is whe there are two-to-oe cotiuous fuctios f : X Y defied o a space X, but give ay such fuctio the image space Y is determied up to a homeomorphism. For example, if f : [0, 1) S 1, where S 1 is the uit circle {p R 2 : p = 1}, is defied by f(x) = exp(4πix), the f is a two-to-oe cotiuous fuctio. Mioduszewski [Mi61] proved that if 2000 Mathematics Subject Classificatio. Primary 54A25. Key words ad phrases. Two-to-Oe maps, Stoe-Čech Compactificatio, Cotiuum Hypothesis, Irreducible maps, Cotiuous Images. 1

2 2 ALAN DOW AND GETA TECHANIE f : [0, 1) Y is a two-to-oe cotiuous fuctio, the Y is homeomorphic to S 1. I this paper we ivestigate the behavior of two-to-oe cotiuous maps defied o N, the remaider βn \ N of the Stoe-Čech compactificatio of the space N of atural umbers. We give partial aswers to questios recetly raised by R. Levy [Le04]. I particular, we show that every twoto-oe cotiuous image of N is homeomorphic to N whe the cotiuum hypothesis (CH) is assumed. There are two-to-oe cotiuous fuctios defied o N. For example, the space N is homeomorphic to N 2, so the projectio map f : N 2 N give by f(x, k) = x is a two-to-oe cotiuous fuctio o N ad the image is N. Such a example would be called trivial. A cotiuous two-to-oe fuctio f : N N is trivial if there is a clope subset C of N such that the restrictios f C ad f (N \C) are homeomorphisms. I [Do04], the first author proved that all maps o N that are two-to-oe are trivial, i the above sese, uder the presece of the Proper Forcig Axiom (PFA). Eric va Douwe [vd93] has also produced a surprisig aswer to a similar questio raised by R. Levy. He showed that the space N, which is a compact space ad very far from beig separable, ca be mapped oto a compact separable space by a two-to-oe cotiuous fuctio. We are cocered with the questio of whether every exactly two-to-oe cotiuous image of N is homeomorphic to N. It is well kow that if a space Y is homeomorphic to N, the Y is a Parovičeko space, that is, a compact zero-dimesioal F -space of weight c which has o isolated poits ad with the property that every oempty

3 TWO TO ONE CONTINUOUS IMAGES OF N 3 G δ -set has ifiite iterior. Therefore, if we are iterested i whether or ot two-to-oe cotiuous images of N are homeomorphic to N, we should ivestigate which of these six properties are satisfied by the spaces which are two-to-oe cotiuous images of N. If Y is a two-to-oe cotiuous image of N, obviously Y is compact sice N is compact ad a cotiuous image of a compact space is compact, ad Y has o isolated poits sice two-to-oe cotiuous maps preserve the property of havig o isolated poits. Roie Levy [Le04] showed that Y has the property that coutable discrete subsets are C - embedded ad Y cotais a copy of N ad so Y has weight c. We iclude his proof for the reader s coveiece. Theorem 1 (Levy). Let X be a space such that every coutable discrete subset of X is C -embedded i X. If f : X Y is a two-to-oe cotiuous fuctio, the every coutable discrete subset of Y is C -embedded i Y. Proof. Let C be a coutable discrete subset of Y. Sice f is two-to-oe ad C is coutable discrete, f 1 (C) is a coutable discrete subset of X. Therefore, f 1 (C) is C -embedded i X. We must show that disjoit subsets of C have disjoit closures i Y. Let A, B C such that A B =. Assume that there exists p Cl Y A Cl Y B. For each a A let f 1 ({a}) = {a 1, a 2 } ad for each b B, let f 1 ({b}) = {b 1, b 2 }. Let A 1 = {a 1 : a A}, A 2 = {a 2 : a A}, B 1 = {b 1 : b B}, B 2 = {b 2 : b B}. These four sets are pairwise disjoit subsets of f 1 (C) ad therefore their closures are also pairwise disjoit sice f 1 (C) is C - embedded. By the cotiuity of f, each of the four sets Cl X A 1, Cl X A 2, Cl X B 1, Cl X B 2 cotais a elemet of the fiber f 1 ({p}).

4 4 ALAN DOW AND GETA TECHANIE Sice these sets are pairwise disjoit, f 1 ({p}) 4. This cotradicts the fact that f is two-to-oe. The Levy asked i the same paper whether Y has the remaiig three properties. We show that Y actually has these three properties uder CH, that is, Y is a zero-dimesioal F -space i which every oempty G δ -set has ifiite iterior. A cosequece the is that two-to-oe cotiuous images of N are homeomorphic to N whe CH is assumed sice N is the oly Parovičeko space uder CH [vm84]. 2. Irreducible Maps A mappig f of X oto Y is irreducible if o proper closed subset of X maps oto Y. Thus, the image of a ope set by a closed irreducible mappig will have iterior. It follows easily from Zor s lemma [Wa74] that if X ad Y are compact spaces ad f is a cotiuous fuctio from X oto Y, the there is a closed subspace F of X such that f F is a irreducible map from F oto Y. Levy [Le04] asked if there exists a irreducible two-tooe cotiuous fuctio whose domai is N. Uder CH, we will show that there is o such fuctio. Notatio. For a map f : X Y ad A X, let J A = f 1 (f[x\a]) A ad for a poit x, x will deote a poit x x such that f(x) = f(x ). Lemma 2. Let f : X Y be a irreducible cotiuous closed map. If A is a ope subset of X, the J A is owhere dese i X. Proof. Suppose that J A is ot owhere dese i X. The It J A is a oempty ope subset of X. The clearly f[a It J A ] f[a] ad J A

5 TWO TO ONE CONTINUOUS IMAGES OF N 5 f 1 (f[x \ A]) sice J A f 1 (f[x \ A]) ad f[x \ A] is closed. Thus f[ J A ] f[x \ A] ad i particular f[a It J A ] f[x \ A]. Therefore, X \ (A It J A ) is a proper closed subset of X sice A It J A oempty ope set cotaied i A, ad f[x \ (A It J A )] = Y sice f[a It J A ] f[a] ad f[a It J A ] f[x \ A]. This is a cotradictio is a sice f is irreducible. The followig result is the mai igrediet i our aalysis of the structure of two-to-oe cotiuous fuctios. Theorem 3 (CH). Let X be a compact space of weight c ad suppose that coutable discrete subsets of X are C -embedded. If f : X K is a twoto-oe cotiuous fuctio ad Z is a closed subset of X such that f Z is irreducible ad maps Z oto K, the for every oempty ope set W K there exists a ope set B i X such that B Z = ad It(f[B]) W. Furthermore, if X is zero-dimesioal, the B ca be chose to be clope. Proof. Let W be a oempty ope subset of K. Seekig a cotradictio, suppose that f[b] W is owhere dese i K for each ope set B X with closure disjoit from Z. For each ope subset B X \ Z, let I B = f 1 (f[b] W ) Z. The I B is owhere dese i Z sice f is closed irreducible ad f[i B ] f[b] W which is owhere dese. Sice f is cotiuous, f 1 (W ) Z has oempty iterior i Z. For each ope subset A f 1 (W ) Z, J A is owhere dese i Z by Lemma 2. We costruct, by iductio, a family { A α : α < ω 1 } which is a filter base of cozero subsets of f 1 (W ) Z such that A α is a sigleto {x} so that α< ω 1

6 6 ALAN DOW AND GETA TECHANIE f 1 (f(x)) = {x}. This yields a cotradictio as f is a exactly two-to-oe fuctio. Let {B α : α < ω 1 } eumerate all the cozero sets B i X such that B Z = ad let { Cα, 0 Cα 1 : α < ω 1 } eumerate all pairs of cozero sets i Z such that Z = Cα 0 Cα. 1 We costruct { A α : α < ω 1 } such that for each α < ω 1 : (1) A β is oempty; β α (2) A α C 0 α or A α C 1 α; ad (3) If α = β + 1, the A α A β \ (J Aβ I Bβ ). We show how to defie the first two cozero sets A 0 ad A 1 : Let x f 1 (W ) Z. The choose a cozero set eighborhood A 0 of x such that A 0 f 1 (W ) Z ad either A 0 C0 0 or A 0 C0. 1 The J A0 ad I B0 are owhere dese i Z, hece A 0 \(J A0 I B0 ). Let x, possibly differet from the previous x, be a poit i A 0 \(J A0 I B0 ). The choose a cozero set eighborhood A 1 of x such that A 1 A 0 \(J A0 I B0 ) ad either A 1 C1 0 or A 1 C1. 1 For each ω we ca defie A i the same maer. Suppose that α ω ad we have costructed the family { A β : β < γ } for all γ < α. If α = β + 1, the A γ by the iductio assumptio. γ β If α is a limit, the the iductio hypothesis (1) esures that G α = A β β<α is ot empty sice Z is compact ad A β A β+1 for each β < α so that A β A β+1. β<α β<α If α is a limit ad x G α = A β, we choose a cozero set eighborhood β<α A α of x such that A α C 0 α if x C 0 α or A α C 1 α otherwise. If α = β + 1 with β ω, we must defie a cozero set A α so that A α A β \ [J Aβ I Bβ ].

7 TWO TO ONE CONTINUOUS IMAGES OF N 7 Let λ be the largest limit ordial less tha α. We eumerate λ {β} by: β = β 0, β 1, β 2, β 3,. We ow cosider the cases whe G α has oempty iterior ad G α is owhere dese. If G α has oempty iterior, the G α \ [J Aβ I Bβ ] sice J Aβ I Bβ is owhere dese. The pick a poit x G α \ [J Aβ I Bβ ] ad a cozero set A α cotaiig x such that A α G α ad either A α Cα 0 or A α Cα. 1 If G α is owhere dese, J Aβ I Bβ G α is owhere dese ad A β \[J Aβ I Bβ G α ]. Let x 0 A β \[J Aβ I Bβ G α ]. The choose i 1 > 1 large eough such that x 0 / A βi1, which we may do sice x 0 / G α. The either x 0 A βi1, x 0 J Aβi1, or x 0 / Z. I ay case, x 0 / A βi1 +1 sice A βi1 +1 A βi1 \ J Aβi1 by costructio. So there is a i 1 > 1 such that β i = β i1 + 1 ad x 0 / A βi. 1 1 Thus x 0, x 0 / G α. Similarly, pick x 1 [A β A β1 A βi1 ]\[J Aβ I Bβ G α ] ad choose i 2 > i 1, i 1 > 1 large eough so that x 1 / A βi2. The x 1 / A βi2 +1 by costructio. So there is a i 2 > i 1, i 1 > 1 such that β i 2 = β i2 + 1 ad x 1 / A βi. Thus x 1, x 1 / G α. Cotiuig this process, for every iteger, 2 pick x [A β A β1 A βi ] \ [J Aβ I Bβ G α ] ad choose a iteger i +1 > i, i > > i 2, i 2 > i 1, i 1 > 1 large eough so that x / A βi+1. The there exists a iteger i +1 > i, i such that x / A βi. Thus x, x / G α. Hece, we get a coutable set {x } {x }. We remark that the x s ad x s ca be chose from some dese subset of f 1 (W ) Z. From the

8 8 ALAN DOW AND GETA TECHANIE costructio of the x s ad x s i > x i, x i f 1 (f[a β A β1 A βi ]) j x j, x j f 1 (f[a β A β1 A βi+2 ]) ad sice f 1 (f[a β A β1 A βk ]) is closed for all k, the set {x } {x } is discrete. Therefore {x } {x } = sice {x } {x } = ad coutable discrete subsets of X are C -embedded. We have f({x } ) = f({x } ) by cotiuity of f ad the fact that f({x } ) = f({x } ). We also have {x } \{x } ad {x } \{x } sice every ifiite subset of a compact set has a limit poit. By the costructio of the x s we see that {x } \{x } G α. If {z Z : z Z} f 1 (W ) is dese i f 1 (W ) Z, we ca choose the x s i f 1 (W ) Z so that we also have {x } \{x } G α. I this case if we choose x {x } \{x }, the x {x } \{x } ad x x sice {x } {x } =. Moreover x J Aβ sice x J Aβ implies x A β which cotradicts x G α A β. We also have x I Bβ, that is, x B β sice x B β for all ad B β is a cozero set. Thus we have foud a x G α such that x J Aβ I Bβ. If {z Z : z Z} f 1 (W ) is owhere dese i f 1 (W ) Z, fid a cozero set A A β f 1 (W ) Z such that f Z is oe-to-oe o all poits of A, that is, f 1 (f[a]) meets Z i A. I this case we ca choose the x s so that {x } A ad hece {x } X \Z. The {x } \{x } A. Choose a x {x } \{x }. The x A ad so x Z; i particular x Z \ A β ad so x J Aβ. It is also true that x I Bβ, that is, x B β sice x B β for all ad B β is a cozero set. Therefore x G α ad x J Aβ I Bβ.

9 TWO TO ONE CONTINUOUS IMAGES OF N 9 We choose a cozero set A α cotaiig x such that A α A β \[J Aβ I Bβ ] ad either A α C 0 α or A α C 1 α. The A α satisfies all the iductio assumptios ad this completes the iductive costructio. But ow A α sice A β A β+1 ad X is compact. α<ω 1 β<ω 1 β<ω 1 Moreover the fact that A α is a sigleto is easily see by the iductio α<ω 1 hypothesis (2). Let A α = {x}. α<ω 1 Claim 1. f 1 (f(x)) = {x}. Proof of Claim. Suppose that for some x x, we have f(x) = f(x ). If x Z, the x / A α for some α < ω 1. This implies x J Aα ad so x / A α+1. This is a cotradictio. If x / Z, the x B α for some α < ω 1. This implies x I Bα ad so x / A α+1. This is also a cotradictio. Therefore f 1 (f(x)) = {x}. This cotradicts f beig exactly two-to-oe fuctio. Moreover the zerodimesioal case follows immediately from the geeral case. If X is zerodimesioal, ad B is a ope set such that B Z =, the there is a clope set disjoit from Z cotaiig B. Corollary 4 (CH). Let X be the Stoe-Čech remaider of a locally compact separable metric space X. If f : X K is a two-to-oe cotiuous fuctio, the f is ot irreducible. I particular, if f : N K or f : R K is two-to-oe ad cotiuous, the f is ot irreducible. Proof. Suppose that f is irreducible. Takig X = Z = X ad W = K i Theorem 3 we get a oempty subset B of the empty set X \ Z.

10 10 ALAN DOW AND GETA TECHANIE Corollary 5 (CH). If f : N K is a two-to-oe cotiuous fuctio, the K is ot ccc. Proof. Let W be a oempty ope subset of K. By Zor s Lemma [Wa74], there is a closed subset Z of N such that f Z : Z K is irreducible. The, by Theorem 3, there exists a oempty clope set B N \Z such that It(f[B]) W sice N is zero-dimesioal. The f B is a closed oe-to-oe fuctio ad so B is homeomorphic to f[b]. Thus, f[b] has o ope ccc subset sice N has o ope ccc subset. Therefore, K is ot ccc. 3. Examples of Notrivial Two-to-Oe Maps A two-to-oe fuctio f : X Y will be called trivial if there exist disjoit clope sets A ad B such that X = A B ad f[a] = f[b] = Y. I [Do04] the first author proved that all fuctios defied o N that are two-to-oe cotiuous are trivial uder PFA. I this sectio we will give some otrivial examples of two-to-oe cotiuous fuctios defied o N whe CH is assumed. A poit is called a P-poit if the family of its eighborhoods is closed uder coutable itersectios. A subset of a space is a P-set if the family of its eighborhoods is closed uder coutable itersectios. CH implies that N has P-poits ad cotais a owhere dese closed P-set which is homeomorphic to N [vm84].

11 TWO TO ONE CONTINUOUS IMAGES OF N 11 Example 1 (CH): We give a example of a otrivial two-to-oe cotiuous fuctio f : N N such that f is locally oe-to-oe at every poit of N except for two P-poits. Cosider two copies of N : N 1, N 2. Let p 1 N 1 ad p 2 N 2 be P-poits. There is a homeomorphism g : N 1 N 2 such that g(p 1 ) = p 2 uder CH [vm84]. The g 1 : N 2 N 1 is also a homeomorphism ad g 1 (p 2 ) = p 1. The free uio of the two copies of N : N 1 N 2 is homeomorphic to N. Let h 1 : N 1 N 2 N 1 N 2 be defied by h 1 = g g 1. The h 1 2 = id. I a similar maer defie h 2 : N 3 N 4 N 3 N 4 so that h 2 = g g 1 ad h 2 2 = id, where N 3 ad N 4 are other copies of N with correspodig P-poits p 3 ad p 4 ad g : N 3 N 4 is a homeomorphism with g(p 3 ) = p 4. The quotiet spaces (N 1 N 2)/p 1 p 2 (N 3 N 4)/p 3 p 4 (N 1 N 4)/p 1 p 4 idetifyig p 1 ad p 2, p 3 ad p 4, p 1 ad p 4, as sigle P-poits i their respective spaces, are homeomorphic to N [vm84]. The free uio (N 1 N 2)/p 1 p 2 (N 3 N 4)/p 3 p 4 is also homeomorphic to N. Now defie f o this space by f : [(N 1 N 2)/p 1 p 2 (N 3 N 4)/p 3 p 4 ] (N 1 N 4)/p 1 p 4 h 1 (x) if x N 2\{p 2 } h 2 (x) if x N 3\{p 3 } f(x) = x if x N 1\{p 1 } N 4\{p 4 } p 1 p 4 if x {p 1 p 2, p 3 p 4 }

12 12 ALAN DOW AND GETA TECHANIE The f is a cotiuous ad exactly two-to-oe fuctio. Moreover, the image of f is homeomorphic to N. We ow itroduce some otatio that will be used i our future discussios about this kid of two-to-oe cotiuous maps. Let X 0 = (N 1 N 2)/p 1 p 2 (N 3 N 4)/p 3 p 4 I 0 = {A X 0 : A = A 0 A 0, A 0, A 0 clope, f[a 0 ] = f[a 0]} So I 0 is a family of clope sets A i X 0 such that A = f 1 (f[a]), i.e., saturated, ad f is locally oe-to-oe o A. Let U 0 deote the uio of all the A s i I 0, ad i this example U 0 = A I 0 A U 0 = X 0 \ {p 1 p 2, p 3 p 4 }. Thus f is locally oeto-oe except at the two P-poits p 1 p 2 ad p 3 p 4. Let X 1 = X 0 \U 0 which is agai for this example give by X 1 = {p 1 p 2, p 3 p 4 } The I 1 is the aalogous set i X 1 but the poits i X 1 are ot i U 0 because as ca be see above f is ot locally oe-to-oe at the poits p 1 p 2 ad p 3 p 4. Usig a similar costructio to Veličković s poset [Ve93], Example 1 ca be doe cosistet with MA+ CH. But MA+ CH is ot by itself eough to do the costructio because of the first author s PFA result [Do04].

13 TWO TO ONE CONTINUOUS IMAGES OF N 13 Example 2 (CH): We give a example of a otrivial two-to-oe cotiuous fuctio f : N N such that f is locally oe-to-oe at every poit of N except for two P-sets. We exted the first example by cosiderig owhere dese closed P-sets istead of P-poits. Cosider two copies of N : N 1, N 2. Let P 1 N 1 ad P 2 N 2 be two differet closed P-sets such that P 1 is homeomorphic to P 2. There is a homeomorphism g 12 : N 1 N 2 such that g 12 (P 1 ) = P 2 uder CH [vm84]. Therefore g 1 12 : N 2 N 1 is also a homeomorphism ad g 1 12 (P 2 ) = P 1. The free uio of the two copies of N : N 1 N 2 is homeomorphic to N. Let h 1 : N 1 N 2 N 1 N 2 be defied by h 1 = g 12 g 1 12 The h 1 (x) x for each x ad h 1 2 = id. I a similar maer defie h 2 : N 3 N 4 N 3 N 4 N 4 so that h 2 = g 34 g 1 34 ad h 2 2 = id, where N 3 ad are other copies of N with correspodig homeomorphic P-sets P 3 ad P 4. The adjuctio spaces N 1 g1 N 2 N 3 g2 N 4 N 1 g3 N 4 where we idetify the P-sets P 1 with P 2, P 3 with P 4, ad P 1 with P 4 are homeomorphic to N [vm84]. The free uio (N 1 g1 N 2) (N 3 g2 N 4) is also homeomorphic to N. Let ϕ : P 2 P 4 be a homeomorphism. Now let us defie f : [(N 1 g1 N 2) (N 3 g2 N 4)] N 1 g3 N 4

14 14 ALAN DOW AND GETA TECHANIE by h 1 (x) if x N 2 \P 2 f(x) = h 2 (x) x if if x N3 \P 3 x N1 \P 1 N4 ϕ(x) if x P 2 The f is a cotiuous ad exactly two-to-oe fuctio ad the image is homeomorphic to N. Let us fid the sets I 0, U 0, X 1, I 1, U 1, ad X 2 for this example which are itroduced i Example 1. X 0 = N 1 g1 N 2 N 3 g2 N 4 I 0 = {A X 0 : A = A 0 A 0, A 0, A 0 clope, f[a 0 ] = f[a 0]} U 0 = A = X 0 \ (P 2 P 4 ) A I 0 X 1 = X 0 \U 0 = P 2 P 4 The fuctio f is ot locally oe-to-oe i the owhere dese closed sets P 2 ad P 4. But f X1 is a cotiuous two-to-oe fuctio. I 1 is the aalogous set i X 1, U 1 = A = X 1, ad X 2 = X 1 \U 1 =. A I 1 Example 3 (CH): We give a example of a otrivial two-to-oe cotiuous fuctio f : N N which is locally oe-to-oe at every poit of N except for two P-sets ad with the property that X 2 ad X 3 =. We kow that N ca be embedded as a owhere dese P-set i N assumig CH [vm84]. Cosider two copies of N : N 5, N 6. Embed N 1 g1 N 2 ad N 3 g2 N 4 i Example 2 as owhere dese P-sets P 5 ad P 6 i N 5 ad N 6, respectively: N 1 g1 N 2 N 5 ad N 3 g2 N 4 N 6

15 TWO TO ONE CONTINUOUS IMAGES OF N 15 I a similar fashio to Example 1, let g : N 5 N 6 be a homeomorphism such that g(p 5 ) = P 6. Therefore g 1 : N 6 N 5 is also a homeomorphism ad g 1 (P 6 ) = P 5. Let h 1 : N 5 N 6 N 5 N 6 be defied by h 1 = g g 1 The h 1 2 = id. Let N 7 N 8 be aother copy of N 5 N 6. Suppose that h 2 : N 7 N 8 N 7 N 6 is defied similarly so that h 2 2 = id. The adjuctio spaces N 5 g5 N 6, N 7 g6 N 8, N 5 g7 N 8 where we idetify the P-sets P 5 with P 6, P 7 with P 8, ad P 5 with P 8 are homeomorphic to N [vm84]. The free uio (N 5 g5 N 6) (N 7 g6 N 8) is also homeomorphic to N. Let ϕ : P 6 P 8 be the two-to-oe fuctio defied i Example 2. Defie f f : [(N 5 g5 N 6) (N 7 g6 N 8)] N 5 g7 N 8 by h 1 (x) if x N 6 \P 6 f(x) = h 2 (x) x if if x N 7\P 7 x N 5 \P 5 N 8\P 8 ϕ(x) if x P 6 P 8 Let us fid the sets I 0, U 0, X 1, I 1, U 1, X 2, I 2, U 2, X 3 i this example. X 0 = (N 5 g5 N 6) (N 7 g6 N 8) I 0 = {A X 0 : A = A 0 A 0, A 0, A 0 clope, f[a 0 ] = f[a 0]} U 0 = A = X 0 \ (P 6 P 8 ) A I 0 X 1 = X 0 \U 0 = P 6 P 8

16 16 ALAN DOW AND GETA TECHANIE The fuctio f is ot locally oe-to-oe i the owhere dese closed sets P 6 ad P 8. Now f X1 is a exactly two-to-oe cotiuous fuctio which is the same as the fuctio i Example 2. Therefore, I 1 = {A X 1 : A = A 1 A 1, A 1, A 1 clope, f[a 1 ] = f[a 1]} U 1 = A = X 1 \ (P 2 P 4 ) A I 1 X 2 = X 1 \U 1 = P 2 P 4 I 2 = {A X 2 : A = A 2 A 1, A 2, A 2 clope, f[a 2 ] = f[a 2]} U 2 = A = X 2 A I 2 X 3 = X 2 \U 2 =. It is clear that we ca cotiue this process for ay fiite umber of steps i the followig sese: If f : N K is a two-to-oe cotiuous fuctio ad X 0 = N, the for each iteger I = {A X : A = A A, A, A clope, f[a ] = f[a ]} U = A I A X +1 = X \U. The for each iteger there is a f so that X while X +1 =.

17 TWO TO ONE CONTINUOUS IMAGES OF N Zero-dimesioal Spaces A space X is called zero-dimesioal if it has a base cosistig of clope sets, that is, if for every poit x X ad for every eighborhood U of x there exists a clope subset C X such that x C U. N is a zero-dimesioal space ad i this sectio we show that every two-to-oe cotiuous image of N is zero-dimesioal uder CH. Suppose that f : N K is a two-to-oe cotiuous fuctio. As i the examples give i sectio 3, let X 0 = N, K 0 = K, I 0 = {A X 0 : A = A 0 A 0, A 0, A 0 clope, f[a 0 ] = f[a 0]}, ad U 0 = A I 0 A. Claim 2. I 0 Proof of claim. By Theorem 3, there is a clope set B N such that f B is oe-to-oe ad Itf[B]. Therefore, f[b] is homeomorphic to B ad hece there is a clope set B Itf[B] ad f 1 [B ] is clope sice f B is oe-to-oe ad it ca be writte as a uio of two disjoit clope sets f 1 [B ] = A 0 A 0 such that f[a 0 ] = f[a 0]. Therefore, f 1 [B ] I 0. This shows I 0 is oempty ad f[u 0 ] is dese i K 0. Now let X 1 = X 0 \U 0 ad K 1 = K 0 \f(u 0 ). The X 1 is a closed subset of X 0. If X 1, the f X1 : X 1 K 1 is a exactly two-to-oe cotiuous

18 18 ALAN DOW AND GETA TECHANIE fuctio. I a similar way as before let I 1 = {A X 1 : A = A 1 A 1, A 1, A 1 clope, f[a 1 ] = f[a 1]} U 1 = A X 2 = X 1 \U 1 K 2 = K 1 \f(u 1 ) A I 1 If X 1, the I 1 by Theorem 3. If X 2, the f X2 : X 2 K 2 is a exactly two-to-oe cotiuous fuctio. Cotiuig i a similar fashio, for each we defie I = {A X : A = A A, A, A clope, f[a ] = f[a ]} U = A I A X +1 = X \ U K +1 = K \ f(u ) The X ω = X ad K ω = K. Recall that we showed i sectio 3 that X may be oempty for ay give atural umber. Therefore, the ext result is quite a surprise. Theorem 6 (CH). X ω = ad K ω =. Proof. Suppose X ω. The I ω where I ω = {A X ω : A = A ω A ω, A ω, A ω clope, f[a ω ] = f[a ω]} Therefore, there exist two oempty disjoit clope sets A ω, A ω X ω such that f[a ω ] = f[a ω]. Sice X ω is compact ad X 0 is zero-dimesioal there are disjoit clope sets B 0, B 0 X 0 such that B 0 X ω = A ω ad B 0 X ω = A ω. Thus X 0 (B 0 B 0) is clope i X 0 ad ) (A ω A ω) f (f(x 1 0 (B 0 B 0)) =

19 TWO TO ONE CONTINUOUS IMAGES OF N 19 by the defiitio of A ω ad A ω. But A ω A ω = = = ( ) B 0 B 0 ( ) B 0 B 0 X ω ( ) X ( ) B 0 B 0 X Therefore f 1 ( f[x 0 \ (B 0 B 0)] ) ( (B 0 B 0) X m ) = for some m. Claim 3. 0 > m such that > 0 f 1 (f[b 0 ]) X B 0. Proof of claim. Otherwise > 0 x, x ( X B 0) \f 1 (f[b 0 ]) such that f(x ) = f(x ). The {x } {x } is a discrete subset of N ad therefore {x } {x } =. Moreover {x }\{x } {x }\{x } X ω B 0 = A ω ad {x }\{x } {x }\{x } is oempty sice every ifiite discrete set i a compact space has a limit poit. But the, there are elemets x {x }\{x } ad x {x }\{x } such that f(x) = f(x ). This is a cotradictio sice f A ω is oe-to-oe. Therefore 0 > m such that > 0 f 1 (f[b 0 ]) X B 0. By symmetry k 0 > m such that > k 0 f 1 (f[b 0 ]) X B 0. Let k = max{k 0, 0 }. The f(b 0 X k+1 ) = f(b 0 X k+1 ). This implies A ω U k+1 ad A ω U k+1. This is a cotradictio sice A ω, A ω X ω X k+1 \ U k+1. Hece X ω = ad K ω =. Lemma 7. If A X 1 is clope with f 1 (f[a]) = A ad U X 0 is ope with A U, the there is a clope set A U i X 0 such that A X 1 = A ad f 1 (f[a ]) = A.

20 20 ALAN DOW AND GETA TECHANIE Proof. Sice A is clope i X 1 ad X 1 is a subspace of X 0, there is a clope set B X 0 such that B X 1 = A ad B U. The f[x 0 \ B] f[a] =. Let A = B\f 1 (f[x 0 \ B]). We ow show that A is the clope subset of X 0 we are lookig for. Clearly A is ope i X 0, A U, A X 1 = A, ad f 1 (f[a ]) = A. It remais to show that A is closed i X 0. This is equivalet to showig that B f 1 (f[x 0 \ B]) is ope. Let x B f 1 (f[x 0 \ B]) ad let x X 0 such that f(x) = f(x ). This implies x, x U 0 = X 0 \ X 1. Therefore, by the defiitio of U 0, there are disjoit clope sets A 0, A 0 U 0 i X 0 such that x A 0, x A 0, ad f[a 0 ] = f[a 0]. Now shrik A 0 ad A 0 to clope sets B 0 ad B 0, respectively, so that x B 0 B, x B 0 f 1 (f[x 0 \ B]), ad f[b 0 ] = f[b 0]. The x B 0 B f 1 (f[x 0 \ B]). Therefore, B f 1 (f[x 0 \ B]) is ope ad A is closed i X 0. Lemma 8. If A X +1 is clope with f 1 (f[a]) = A ad U X is ope with A U, the there is a clope set A U i X such that A X +1 = A ad f 1 (f[a ]) = A. Proof. The proof is similar to the proof of Lemma 7 with X +1 ad X playig the roles of X 1 ad X 0, respectively. Lemma 9. If A X is clope with f 1 (f[a]) = A ad U X 0 is ope with A U, the there is a clope set A U i X 0 such that A X = A ad f 1 (f[a ]) = A. Proof. This follows from Lemma 7 ad Lemma 8 by iductio. Theorem 10 (CH). If f : N K is a two-to-oe cotiuous fuctio, the K is zero-dimesioal.

21 TWO TO ONE CONTINUOUS IMAGES OF N 21 Proof. Let y V where V is a ope subset of K. The y K \K +1 sice K ω = by Theorem 6. This implies y f(u ) = K \ K +1. Therefore, y f[a ] f(u ) = K \ K +1 for some clope set A X such that f[a ] is clope ad f A is oe-to-oe. This is by the defiitio of U. The f[a ] is homeomorphic to A ad so there is a clope set B It f[a ] cotaiig y. Shrik B so that y B V K. Let A = f 1 (B) ad U = f 1 (V ). The A X is clope with f 1 (f[a]) = A ad U X 0 is ope with A U. By Lemma 7, A U clope i X 0 such that A X = A ad f 1 (f[a ]) = A. The y f[a ] V ad f[a ] is clope i K sice f 1 (f[a ]) = A. Hece K is zero-dimesioal. 5. F -spaces A space is called a F-space if every pair of disjoit cozero subsets are completely separated. It is well kow that N is a F -space [Wa74] ad i this sectio we show that every two-to-oe cotiuous image of N is also a F -space uder CH. Theorem 11 (CH). If f : N K is a two-to-oe cotiuous fuctio, K is a F-space. Proof. Let C 1 ad C 2 be two disjoit cozero sets i K. The f 1 (C 1 ) ad f 1 (C 2 ) are disjoit cozero sets i N. Sice N is a F -space we have f 1 (C 1 ) f 1 (C 2 ) =. We must show that C 1 C 2 =. It is sufficiet to show that for ay y C 1 there are two elemets x, x f 1 (C 1 ) such that f(x) = y = f(x ). This shows that y / C 2. Otherwise, if y

22 22 ALAN DOW AND GETA TECHANIE [ ] C 2 f f 1 (C 2 ), there exists a x f 1 (C 2 ) such that f(x ) = y ad x x, x sice f 1 (C 1 ) f 1 (C 2 ) = ad x, x f 1 (C 1 ). So three differet poits x, x, x mapped to y. This is a cotradictio to the fact that the fuctio f is exactly two-to-oe. Let y C 1. The y C 1 f[f 1 (C 1 )] sice C 1 = f[f 1 (C 1 )] f[f 1 (C 1 )]. This implies there exists a x f 1 (C 1 ) such that f(x) = y. By Theorem 6 K ω = ad so y K \ K +1 for some iteger. Let m be maximal such that y C 1 K m. The y / C 1 K m+1 ad so there is a cozero set C y K m such that y C y ad C y C 1 K m+1 =. Thus C y C 1 K m+1 =. Therefore, without loss of geerality, we ca assume that C 1 K m+1 = ad C 1 is a cozero set i K m sice we ca take C 1 to be the cozero set C y C 1. The f 1 (C 1 ) is a cozero set i X m ad f 1 (C 1 ) X m+1 =, that is, f 1 (C 1 ) U m where U m is defied as i sectio 4 by U m = ad A I m A I m = {A X m : A = A m A m, A m, A m clope, f[a m ] = f[a m]} Sice f 1 (C 1 ) is a cozero set i X m ad X m is compact zero-dimesioal F -space, it ca be writte as a coutable uio of disjoit clope sets i such a way that f 1 (C 1 ) = [ ] A A where each A ad A are disjoit clope sets i X m ad f[a ] = f[a ]. Therefore, f 1 (C 1 ) ca be writte as a uio of two disjoit sets ( ) ( f 1 (C 1 ) = A A )

23 TWO TO ONE CONTINUOUS IMAGES OF N 23 ad by the defiitio of A ad A we get [ ] [ f A = f Thus f 1 (C 1 ) = The sets A ad A A A ]. A. are cozero sets sice a coutable uio of clope sets is a cozero set ad they are disjoit by costructio. Therefore, sice X m is a F -space we get A A =. Now sice x f 1 (C 1 ) = A we assume, without loss of geerality, that A x A ad x / A. By cotiuity of f ad the fact that [ ] [ f A = f A ] we get ad y C 1 f [ ] [ f A = f [ [ ] f 1 (C 1 ) = f A ] A ] = f [ A ].

24 24 ALAN DOW AND GETA TECHANIE Therefore, there exists a x Now x x because A such that f(x ) = y. x A, x A, ad A A =. Thus, there are two differet poits x, x f 1 (C 1 ) such that f(x) = y = f(x ) Hece K is a F -space. 6. Noempty G δ -sets The itersectio of coutably may ope sets is called a G δ -set. Noempty G δ -sets o N have oempty iteriors. I this sectio we prove that twoto-oe cotiuous images of N have the same property. Theorem 12 (CH). If f : N K is a two-to-oe cotiuous fuctio, the oempty G δ -sets i K have oempty iterior. Proof. Suppose that {b : ω} is a descedig sequece of clope subsets of K with b 0 = K. It suffices to deal with clope sets sice we have show that K is zero-dimesioal. Assume to the cotrary that b is owhere dese. Let Z N be such that f Z is irreducible. This is possible by Zor s Lemma.

25 TWO TO ONE CONTINUOUS IMAGES OF N 25 For each let a = f 1 (b ) Z. The {a : ω} is a descedig sequece of clope subsets of Z ad a 0 = Z. Therefore, by Theorem 3, for each pick clope sets e a \ a +1 ad e N \ Z such that f[e ] = f[e ]. The, i Z, (a \ (e a +1 ) e = sice Z is a F -space. Sice we assumed dese ad so b is owhere dese i K, f 1 ( b ) Z is owhere Z = (a \ (e a +1 ) e. Thus e is clope i Z. Clearly (a \ a +1 ) e = sice Z is closed ad e N \Z. Let us show that (a \a +1 ) e =. For each, f 1 (K\b ) is clope i N sice b is clope ad f is cotiuous. By costructio e m m f 1 (K \ b ) = ad e m f 1 (K b ) is clope i N sice it is a m ω fiite itersectio of clope sets. Because a \ a +1 f 1 (K \ b +1 ) ad [a \ (a +1 e )] e = ω e = sice we have (a \ a +1 ) N is a F -space. Therefore e e =. Thus (a \ a +1 ) e = ad hece f is a two-to-oe fuctio o e e. [ ] [ ] [ ] [ Now sice f e = f e we have K = f e f Z \ ] e [ ] [ ad f e f Z \ ] [ ] e =. Therefore f e K is clope i K. But the [ ] ) f (f 1 e = e e

26 26 ALAN DOW AND GETA TECHANIE is clope. This is a cotradictio sice e e is ot clope by the fact that i N oempty G δ -sets have oempty iterior. If f : N K is a two-to-oe cotiuous fuctio, Levy [Le04] proved that coutable discrete subsets of K are C -embedded ad the weight of K is c. This completes everythig eeded to prove that a two-to-oe cotiuous image of N is N up to a homeomorphism. Corollary 13 (CH). If f : N K is a two-to-oe cotiuous fuctio, the K is homeomorphic to N. Proof. Follows from Theorem 1, 10, 11, ad 12. Ope Problems Our results are all uder the set theoretic assumptio CH. Is it possible to elimiate CH? I particular, if f : N K is a two-to-oe cotiuous fuctio: (1) Is it true that f is ot irreducible? (2) Is K homeomorphic to N? (3) Is every coutable subset of K C -embedded? (4) Ca K be separable or ccc? (Levy questio [vd93]) (5) If f is -to-oe cotiuous with > 2, is K homeomorphic to N uder CH or ZFC?

27 TWO TO ONE CONTINUOUS IMAGES OF N 27 Refereces [vd93] Eric K. va Douwe, Applicatios of Maximal Topologies, Topology ad Its Appl. 51 (1993), o. 2, [Do04] Ala Dow, Two to Oe Images ad PFA, Preprit, [Ha39] O.G. Harrold, The o-existece of a certai type of cotiuous trasformatio, Duke Math. J. 5, (1939) [He86] J. Heath, Every exactly 2-to-1 fuctio o the reals has a ifiite set of discotiuities, Proc. Amer. Math. Soc. 98 (1986), o. 2, [Le04] Roie Levy, The Weight of Certai Images of ω, Preprit, [vm84] Ja va Mill, A Itroductio to βω, Hadbook of Set-Theoretic Topology (K.Kue ad J.E. Vaugha, eds), Elsevier Sciece Publishers BV, North-Hollad, Amsterdam, 1984, pp [Mi61] J. Mioduszewski, O Two-to-Oe Cotiuous Fuctios, Rozprawy Matematycze 24, Warszawa, 1961, 43 pp. [Ve93] Boba Veličković, OCA ad Automorphisms of P (ω)/f i, Topology ad Its Appl. 49 (1993), [Wa74] Russell C. Walker, The Stoe-Čech Compactificatio, Spriger-Verlag, New York, 1974, Ergebisse der Mathematik ud ihrer Grezgebiete, Bad 83. Departmet of Mathematics, Uiversity of North Carolia at Charlotte, 9201 Uiversity City Blvd., Charlotte, NC address: adow@ucc.edu URL: adow Departmet of Mathematics, Uiversity of North Carolia at Charlotte, 9201 Uiversity City Blvd., Charlotte, NC address: gtechai@ucc.edu URL: gtechai