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Central European Journal of Mathematics

The closures of arithmetic progressions

in the common division topology

on the set of positive integers

Research Article

Paulina Szczuka1∗

1 Kazimierz Wielki University, Institute of Mathematics, Pl. Weyssenhoffa 11, 85-072 Bydgoszcz, Poland

Received 25 September 2013; accepted 7 January 2014

Abstract: In this paper we characterize the closures of arithmetic progressions in the topology T on the set of positive

integers with the base consisting of arithmetic progressions {an+ b} such that if the prime number p is a factor

of a, then it is also a factor of b. The topologyT is called the common division topology.

MSC: 54A05, 11B25, 11A41, 11A05

Keywords: The common division topology • Closures • Arithmetic progressions © Versita Sp. z o.o.

1.

Preliminaries

The letters N, N0 and P denote the sets of positive integers, non-negative integers and primes, respectively. For a set

A we use the symbol cl A to denote the closure of A. The symbol Θ(a) denotes the set of all prime factors of a ∈ N. For all a, b ∈ N, we use (a, b) and lcm (a, b) to denote the greatest common divisor of a and b and the least common

multiple of a and b, respectively. Moreover, for all a, b ∈ N, the symbols {an + b} and {an} stand for the infinite

arithmetic progressions: {an+ b} df = a · N0+ b and {an} df = a · N.

Hence, clearly, {an} = {an + a}. For the basic results and notions concerning topology and number theory we refer

the reader to the monographs of Kelley [4] and LeVeque [6], respectively.

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2.

Introduction

In 1955 Furstenberg [2] defined the base of a topology on the set of integers by means of all arithmetic progressions

and gave anelegant topological proof of the infinitude of primes. In1959 Golomb [3] presented a similar proof of

the infinitude of primes using a topology D on N with the base BG = {an+ b} : (a, b) = 1

defined in 1953 by

Brown [1]. Ten years later Kirch [5] defined a topology D

0

on N, weaker than Golomb’s topology D, with the base BK = {{an + b} : (a, b) = 1, a is square-free}. Both topologies D and D

0

are Hausdorff, the set N is connected in

these topologies and locally connected in the topology D

0

, but it is not locally connected in the topology D (see [3,5]).

Moreover, the set N is semiregular in the stronger topology D and it is not semiregular in the weaker topology D

0

(see [10]).

In 1993 Rizza [7] introduced the division topology T

0

on N as follows: for X ⊂ N he put

g(X ) = cl X = [

x∈X

D(x ), where D(x ) = {y ∈ N : y | x }.

The mapping g forms a topology T

0

on N. It is easy to see that the family B

0

= {{an}} is a basis for this topology.

In [9] the author defined the common division topology T on N, stronger than the division topology T

0

, with the base B = {{an + b} : Θ(a) ⊂ Θ(b)}. Both topologies T and T

0

are T0 and they are not T1, the set N is connected in these topologies and locally connected in the topology T

0

, but it is not locally connected in the topology T (see [7,9]).

Moreover, the set N is semiregular in the stronger topology T and it is not semiregular in the weaker topology T

0

(see [10]).

Since 2010 the author has examined properties of arithmetic progressions in the above four topologies. It was already

shown that the base of Golomb’s topology D consists of all arithmetic progressions that are connected in the common

division topology T, and conversely, all arithmetic progressions connected in T form a basis for D (see [9]).

More-over, it turned out that all arithmetic progressions are connected in topologies D

0

and T

0

(see [8, Theorem 3.5] and

[9, Theorem 4.1], respectively). Recently the author gave a characterization of regular open arithmetic progressions in

these topologies (see [10]).

In this paper we continue studies concerning properties of arithmetic progressions, namely, we characterize closures

of arithmetic progressions in the common division topology T on N. From now on we will only deal with the common

division topology and to simplify the notation the symbol T will be omitted.

3.

Main results

We start with two simple technical lemmas.

Lemma 3.1.

Assume that U is an open set. If c ∈ U, then there is an arithmetic progression {an+ c} ∈ B such that {an + c} ⊂ U .

Proof. Let c ∈ U . Since the set U is open, there is an arithmetic progression {an + b} ∈ B such that c ∈ {an + b} ⊂

U and Θ(a) ⊂ Θ(b). So, {an + c} ⊂ {an + b} ⊂ U and Θ(a) ⊂ Θ(c). This implies that {an + c} ∈ B.

Lemma 3.2.

If b1≡ b(mod a), then cl {an + b} = cl {an + b1}.

Proof. Without loss of generality we can assume that b1< b. Since {an + b} ⊂ {an + b1}, we have cl {an + b} ⊂ cl {an + b1}. So, it is sufficient to show the opposite implication.

Let x ∈ cl {an + b1}. Fix an open set U with x ∈ U . By Lemma3.1, there is a basic arithmetic progression {cn + x } ⊂ U .

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two infinite arithmetic progressions is an infinite arithmetic progression, we can conclude that the set {cn + x } ∩ {an + b1}

is infinite. Simultaneously, the set {an + b1} \ {an+ b} is finite, which implies

∅ 6= {an + b} ∩ {cn + x } ⊂ {an + b} ∩ U .

This proves that x ∈ cl {an + b}.

The proof of next remark is evident.

Remark 3.3.

cl {n + b} = N for each b ∈ N.

From now on in all theorems of this paper we assume a > 1.

Theorem 3.4. Assume p ∈P, k ∈ N and b1≤ p k. If b 1≡ b(mod p k ), then cl { p kn + b} = { p kn + b1} ∪(N \ { pn}).In particular, (i) the arithmetic progression {2n + 1} is closed,

(ii) cl { pn} = N for each p ∈ P, and

(iii) if the arithmetic progression {pn+ b} is open, then cl { pn + b} = N for each p ∈ P.

Proof. First we will show that cl { p

kn

+ b} ⊂ {p

kn

+ b1} ∪(N \ { pn}). Using the assumptions b1 ≤ p

k and b1≡ b(mod p k ), we obtain {pkn+ b} ⊂ { p k n+ b1} ⊂ {p k n+ b1} ∪(N \ { pn}).

If (p, b) = 1, then (p, b1) = 1, too. Hence { p

kn

+ b1} ⊂ N \ {pn}and the set N \ { pn} = { p

kn

+ b1} ∪(N \ { pn}) is closed. This proves that cl{ p

kn

+ b} ⊂ { p

kn

+ b1} ∪(N \ { pn}). So, we can assume p | b. Then, obviously, p | b1, whence

{pkn + b1} ⊂ {pn}. Since {pn} \ {pkn+ b1} = pk−1 [ i=1 {pkn+ ip} \ { p k n+ b1} = [ i∈{1,...,pk−1} \ {b 1} {pkn+ ip}

and all arithmetic progressions { p

kn

+ ip} are open, the set { p

kn + b1} ∪(N \ { pn}) = N \ ({ pn} \ { p kn + b1}) is closed. Consequently, cl { p kn + b} ⊂ { p kn + b1} ∪(N \ { pn}). Now we will show the opposite inclusion. Let x ∈ { p

kn

+ b1} ∪(N \ { pn}). We consider two cases.

Case 1: x ∈ {pkn + b1}. Since b1≡ b(mod p k ), by Lemma3.2, cl { p kn + b} = cl { p kn + b1}. So, x ∈ { p kn + b1} ⊂ cl { p kn + b1}= cl { p kn + b}. Case 2: x ∈ N \ {pn}= S

d∈{1,...,p−1}{pn+ d}. Then x ∈ { pn + d} for some d ∈ {1, . . . , p − 1}. Fix an open set U

such that x ∈ U . By Lemma3.1, there is an arithmetic progression {cn + x } ∈ B with {cn + x } ⊂ U and Θ(c) ⊂ Θ(x ).

Since x ∈ N \ { pn}, we have (p, x ) = 1. So, (p, c) = 1, too. Using the Chinese Remainder Theorem (CRT), we obtain

∅ 6= {cn + x } ∩ { p k n+ b} ⊂ U ∩ { p k n+ b}, whence x ∈ cl { p kn + b}.

Finally, observe that conditions (i) and (ii) are evident. Moreover, since the arithmetic progression { pn + b} is open,

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Theorem 3.5. Let a= p α 1 1 . . . p αk

k be the prime power factorization of a. Thencl {an + b} = Tk

i=1cl p

αi

i n+ b .

Proof. First observe

{an+ b} = k \ i=1  pαi i n+ b . Hence cl {an + b} ⊂ Tk i=1cl p αi i n+ b .

Now we will show the opposite inclusion. Assume x ∈ Tk i=1cl p αi i n+ b . Then, by Theorem3.4, x ∈ Tk i=1  p αi i n+ bi ∪(N \ { pin})  , where bi ≡ b(mod p αi i ) and bi ≤ p αi

i for each i ∈ {1, . . . , k }. Fix an open set U such that x ∈ U . By Lemma3.1, there is an arithmetic progression {cn + x } ∈ B with {cn + x } ⊂ U . Hence Θ(c) ⊂ Θ(x ). We consider three

cases.

Case 1: x ∈Tk i=1 p

αi

i n+ bi . By CRT, there is exactly one s ∈ N such that 1 ≤ s ≤ p

α 1 1 . . . p αk k and k \ i=1 {pαi i n+ bi}=  1 1 . . . p αk k  n+ s = {an + s}. Since bi≡ b(mod p αi i ) and bi≤ p αi

i for each i ∈ {1, . . . , k }, we have p

αi i n+ b ⊂ p αi i n+ bi for each i ∈ {1, . . . , k }. Hence {an+ b} = k \ i=1  pαi in+ b ⊂ k \ i=1  pαi i n+ bi = {an + s}.

So, s ≡ b (mod a). By Lemma3.2, we obtain that cl {an + b} = cl {an + s}. Consequently, x ∈ {an + s} ⊂ cl {an + s} =

cl {an + b}. Case 2: x ∈Tk i=1(N \ { pin}). Since Tk i=1(N \ { pin}) = N \ Sk i=1{pin}, we have x∈/ Sk

i=1{pin}. So, (pi, x) = 1 for each

i ∈ {1, . . . , k }. Hence (pi, c) = 1 for each i ∈ {1, . . . , k }, which implies (a, c) = 1. By CRT,

∅ 6= {cn + x } ∩ {an + b} ⊂ U ∩ {an + b}.

Consequently, x ∈ cl {an + b}.

Case 3: There are a number r ∈ {1, . . . , k − 1} and a permutation {σ1, . . . , σk} of the set {1, . . . , k } such that x Tr

i=1 p

ασi

σi n+ bσi ∩ T

k

i=r+1(N \ { pσin}). By CRT, there is exactly one s ∈ N such that 1 ≤ s ≤ p

ασ 1 σ 1 . . . p ασr σr and r \ i=1  pασi σi n+ bσi =  pασ1 σ 1 . . . p ασr σr n+ s . So, x ∈  pασ1 σ 1 . . . p ασr σr  n+ s ∩ N \ k [ i=r+1 {pσin} ! . Define a1= p ασ 1 σ 1 . . . p ασr σr and a2 = p ασr +1 σr+1 . . . p ασk

σk . Then (a1, a2) = 1. Moreover, (pσi, x) = 1 for each i ∈ {r + 1, . . . , k }. Hence (pσi, c) = 1 for each i ∈ {r + 1, . . . , k }, which implies (a2, c) = 1. Since bσi≡ b mod p

ασi σi  and bσi≤ p ασi σi for each i ∈ {1, . . . , r}, we obtain that p ασi σi n+ b ⊂ p ασi σi + bσi

for each i ∈ {1, . . . , r}. So,

{a1n+ b} = r \ i=1  pασi σi n+ b ⊂ r \ i=1  pασi σi n+ bσi = {a1n+ s}.

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Hence

{a1n+ s} ∩ {a2n+ b} = {an + b}. (1)

Since x ∈ {a1n+ s}, we have {a1n+ s} ∩ {cn + x } 6= ∅. It is known that nonempty intersection of two infinite arithmetic progressions is an infinite arithmetic progression. Therefore

{a

1n+ s} ∩ {cn + x } = {dn + e}, where d= lcm (c, a1).

Moreover, if (c, a2) = 1 and (a1, a2) = 1, then (d, a2) = 1. So, by CRT and condition (1), we obtain

∅ 6= {dn + e} ∩ {a2n+ b} = {a1n+ s} ∩ {cn + x } ∩ {a2n+ b} = {cn + x } ∩ {an + b} ⊂ U ∩ {an + b}. Consequently, x ∈ cl {an + b}. Theorem 3.6. Let a= p α 1 1 . . . p αk

k be the prime power factorization of a. Define

A=l ≤ a: (pi, l) = 1 or l ≡ b mod p

αi

i  for each i ∈ {1, . . . , k } .

Then cl{an + b} = S

l∈A{an+ l}. In particular, if a is square-free and the arithmetic progression {an+ b} is open,

thencl{an + b} = N.

Proof. First assume x ∈ cl{an + b}. By Theorems3.5and3.4, respectively,

cl{an + b} = k \ i=1 cl p αi i n+ b = k \ i=1  pαi i n+ bi ∪(N \ { pin}), where bi≡ b(mod p αi i ) and bi≤ p αi

i for each i ∈ {1, . . . , k }.We consider three cases.

Case 1: x ∈Tk

i=1 p

αi

i n+ bi . By CRT, there is exactly one l ∈ N such that 1 ≤ l ≤ p

α 1 1 . . . p αk k and k \ i=1  pαi i n+ bi =  1 1 . . . p αk k  n+ l = {an + l}. Since bi≡ b mod p αi i  and bi≤ p αi

i for each i ∈ {1, . . . , k }, we have p

αi in+ b ⊂ p αi i n+ bi for each i ∈ {1, . . . , k }. Hence {an+ b} = k \ i=1  pαi in+ b ⊂ k \ i=1  pαi i n+ bi = {an + l},

which proves l ≡ b (mod a). Consequently, l ≡ b mod p

αi

i



for each i ∈ {1, . . . , k }. Since l ≤ a, we obtain that l ∈ A,

whence x ∈ S l∈A{an+ l}. Case 2: x ∈Tk i=1(N \ { pin}). Observe k \ i=1 (N \ { pin}) = k \ i=1 pi−1 [ d=1 pαi−1 i 1 [ t=0  pαi in+ (pit+ d) .

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So, for each i ∈ {1, . . . , k } there are di∈ {1, . . . , pi−1} and ti∈  0, . . . , p αi−1 1 such that x ∈ p αi i n+ (piti+ di) . This implies x ∈ Tk i=1 p αi i n+ (piti+ di)

.By CRT, there is exactly one l ∈ N such that 1 ≤ l ≤ p

α 1 1 . . . p αk k and k \ i=1  pαi i n+ (piti+ di) =  1 1 . . . p αk k  n+ l = {an + l}.

Moreover, since di< pifor each i ∈ {1, . . . , k }, we have (di, pi) = 1 for each i ∈ {1, . . . , k }. Therefore, (piti+ di, pi) = 1 for each i ∈ {1, . . . , k } and finally, (l, pi) = 1 for each i ∈ {1, . . . , k }. This proves that l ∈ A, whence x ∈

S

l∈A{an+ l}.

Case 3: There are a number r ∈ {1, . . . , k − 1} and a permutation {σ1, . . . , σk} of the set {1, . . . , k } such that x Tr

i=1 p

ασi

σi n+ bσi ∩ T

k

i=r+1(N \ { pσin}). By CRT, there is exactly one s ∈ N such that1 ≤ s ≤ p

ασ 1 σ 1 . . . p ασr σr and Tr i=1 p ασi σi n+ bσi =  pασ1 σ 1 . . . p ασr σr  n+ s . Moreover, k \ i=r+1 (N \ { pσin}) = k \ i=r+1 pσi1 [ d=1 pασi −1σi 1 [ t=0  pασi σi n+ (pσit+ d) .

So, for each i ∈ {r + 1, . . . , k } there are di ∈ {1, . . . , pσi−1} and ti  0, . . . , p ασi1 1 such that x ∈ p ασi σi n+ (pσiti+ di) . Therefore, x ∈ pασ1 σ 1 . . . p ασr σr  n+ s ∩ k \ i=r+1  pασi σi n+ (pσiti+ di) .

By CRT, there is exactly one z ∈ N such that 1 ≤ z ≤ p

ασr +1 σr+1 . . . p ασk σk and x ∈  pασ1 σ 1 . . . p ασr σr  n+ s ∩  p ασr +1 σr+1 . . . p ασk σk  n+ z .

Now, using once more CRT we obtain that there is exactly one positive integer l ≤ a such that x ∈ {an + l}. Additionally,

since (di, pσi) = 1 for each i ∈ {r + 1, . . . , k }, we have (pσiti+ di, pσi) = 1 for each i ∈ {r + 1, . . . , k } and finally, (pσr+1. . . pσk, z) = 1.So, it is easy to see that

l ≡ s mod p ασ 1 σ 1 . . . p ασr σr  and (pσr+1. . . pσk, l) = 1. (2) Since bσi ≡ b mod p ασi σi  and bσi ≤ p ασi

σi for each i ∈ {1, . . . , r}, we have  p

ασi σi n+ b  pασi σi n+ bσi for each i ∈ {1, . . . , r}. Hence  pασ1 σ 1 . . . p ασr σr  n+ b = r \ i=1  pασi σi n+ b ⊂ r \ i=1  pασi σi n+ bσi =  pασ1 σ 1 . . . p ασr σr  n+ s , which implies s ≡ b mod p ασ 1 σ 1 . . . p ασr σr . (3)

By conditions (2) and (3), l ≡ b mod p

ασ 1 σ 1 . . . p ασr σr  , whence l ≡ b mod p ασi σi 

for each i ∈ {1, . . . , r}. Moreover, by (2),

(pσi, l) = 1 for each i ∈ {r + 1, . . . , k }. Consequently, l ∈ A, whence x ∈ S

l∈A{an+ l}. This completes the first part of the proof.

Now we will show the opposite inclusion. Assume x ∈ S

l∈A{an+ l}. Then x ∈ {an + l} for some l ∈ A. Observe

{an+ l} =  1 1 . . . p αk k  n+ l = k \ i=1  pαi i n+ l . (4)

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We will show  pαi i n+ l ⊂cl p αi i n+ b for each i ∈ {1, . . . , k }. (5)

Fix i ∈ {1, . . . , k }. Condition l ∈ A implies (pi, l) = 1 or l ≡ b mod p

αi i  . If (pi, l) = 1, then p αi i n+ l ⊂ N \ {pin}. By Theorem3.4, N \ { pin} ⊂cl p αi i n+ b

, which proves that p

αi i n+ l ⊂cl p αi i n+ b . If l ≡ b mod p αi i  , then by Lemma3.2, cl p αi i n+ b = cl pαi i n+ l . Therefore, p αi i n+ l ⊂cl p αi in+ b

, which completes the proof of (5). So,

using (4), (5), and Theorem3.5we obtain

x ∈ {an+ l} = k \ i=1  pαi in+ l ⊂ k \ i=1 cl p αi in+ b = cl {an + b}.

Finally, observe that if a is square-free, then a = p1. . . pk. Since

{an+ b} =

k

\

i=1

{pin+ b}

and {an + b} is open, { pin+ b} is also open for each i ∈ {1, . . . , k }. So, Theorem3.5and condition (iii) of Theorem3.4 imply cl {an + b} = k \ i=1 cl { pin+ b} = N.

This completes the proof.

Acknowledgements

Research is supported by Kazimierz Wielki University.

References

[1] Brown M., A countable connected Hausdorff space, In: Cohen L.M., The April Meeting in New York, Bull. Amer.

Math. Soc., 1953, 59(4), 367

[2] Furstenberg H., On the Infinitude of primes, Amer. Math. Monthly, 1955, 62(5), 353

[3] Golomb S.W., A connected topology for the integers, Amer. Math. Monthly, 1959, 66(8), 663–665

[4] Kelley J.L., General Topology, Grad. Texts in Math., 27, Springer, New York-Berlin, 1975

[5] Kirch A.M., A countable, connected, locally connected Hausdorff space, Amer. Math. Monthly, 1969, 76(2), 169–171

[6] LeVeque W.J., Topics in Number Theory, I–II, Dover, Mineola, 2002

[7] Rizza G.B., A topology for the set of nonnegatlve integers, Riv. Mat. Univ. Parma, 1993, 2, 179–185

[8] Szczuka P., The connectedness of arithmetic progressions in Furstenberg’s, Golomb’s, and Kirch’s topologies,

Demon-stratio Math., 2010, 43(4), 899–909

[9] Szczuka P., Connections between connected topological spaces on the set of positive integers, Cent. Eur. J. Math.,

2013, 11(5), 876–881

[10] Szczuka P., Regular open arithmetic progressions in connected topological spaces on the set of positive integers,

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