Arithmetic progression topologies

<h2 id="construction">Construction</h2>
Two-sided arithmetic progressions in 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
 are subsets of the form

a
        
          Z
        
        +
        b
        :=
        {
        a
        n
        +
        b
        :
        n
        ∈
        
          Z
        
        }
        ,
      
    
    {\displaystyle a\mathbb {Z} +b:=\{an+b:n\in \mathbb {Z} \},}

where 
 
 
 
 a
 ,
 b
 ∈
 
 Z
 
 
 
 {\displaystyle a,b\in \mathbb {Z} }
 
 and 
 
 
 
 a
 >
 0.
 
 
 {\displaystyle a>0.}
 
 The intersection of two such arithmetic progressions is either empty, or is another arithmetic progression of the same form:

(
        a
        
          Z
        
        +
        b
        )
        ∩
        (
        c
        
          Z
        
        +
        b
        )
        =
        lcm
        ⁡
        (
        a
        ,
        c
        )
        
          Z
        
        +
        b
        ,
      
    
    {\displaystyle (a\mathbb {Z} +b)\cap (c\mathbb {Z} +b)=\operatorname {lcm} (a,c)\mathbb {Z} +b,}

where 
 
 
 
 lcm
 ⁡
 (
 a
 ,
 c
 )
 
 
 {\displaystyle \operatorname {lcm} (a,c)}
 
 is the <a href="/facts/Least_common_multiple/AX1jIUnw">least common multiple</a> of 
 
 
 
 a
 
 
 {\displaystyle a}
 
 and 
 
 
 
 c
 .
 
 
 {\displaystyle c.}
 
<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
Similarly, one-sided arithmetic progressions in 
 
 
 
 
 
 Z
 
 
 >
 0
 
 
 =
 {
 1
 ,
 2
 ,
 .
 .
 .
 }
 
 
 {\displaystyle \mathbb {Z} _{>0}=\{1,2,...\}}
 
 are subsets of the form

a
        
          N
        
        +
        b
        :=
        {
        a
        n
        +
        b
        :
        n
        ∈
        
          N
        
        }
        =
        {
        b
        ,
        a
        +
        b
        ,
        2
        a
        +
        b
        ,
        .
        .
        .
        }
        ,
      
    
    {\displaystyle a\mathbb {N} +b:=\{an+b:n\in \mathbb {N} \}=\{b,a+b,2a+b,...\},}

with 
 
 
 
 
 N
 
 =
 {
 0
 ,
 1
 ,
 2
 ,
 .
 .
 .
 }
 
 
 {\displaystyle \mathbb {N} =\{0,1,2,...\}}
 
 and 
 
 
 
 a
 ,
 b
 >
 0
 
 
 {\displaystyle a,b>0}
 
. The intersection of two such arithmetic progressions is either empty, or is another arithmetic progression of the same form:

(
        a
        
          N
        
        +
        b
        )
        ∩
        (
        c
        
          N
        
        +
        d
        )
        =
        lcm
        ⁡
        (
        a
        ,
        c
        )
        
          N
        
        +
        q
        ,
      
    
    {\displaystyle (a\mathbb {N} +b)\cap (c\mathbb {N} +d)=\operatorname {lcm} (a,c)\mathbb {N} +q,}

with 
 
 
 
 q
 
 
 {\displaystyle q}
 
 equal to the smallest element in the intersection.

This shows that every nonempty intersection of a finite number of arithmetic progressions is again an arithmetic progression. One can then define a topology on 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
 or 
 
 
 
 
 
 Z
 
 
 >
 0
 
 
 
 
 {\displaystyle \mathbb {Z} _{>0}}
 
 by choosing a collection 
 
 
 
 
 
 B
 
 
 
 
 {\displaystyle {\mathcal {B}}}
 
 of arithmetic progressions, declaring all elements of 
 
 
 
 
 
 B
 
 
 
 
 {\displaystyle {\mathcal {B}}}
 
 to be open sets, and taking the topology generated by those. If any nonempty intersection of two elements of 
 
 
 
 
 
 B
 
 
 
 
 {\displaystyle {\mathcal {B}}}
 
 is again an element of 
 
 
 
 
 
 B
 
 
 
 
 {\displaystyle {\mathcal {B}}}
 
, the collection 
 
 
 
 
 
 B
 
 
 
 
 {\displaystyle {\mathcal {B}}}
 
 will be a <a href="/facts/Base_(topology)/8EDSDVPD">base</a> for the topology. In general, it will be a <a href="/facts/Subbase/CoQAlDFJ">subbase</a> for the topology, and the set of all arithmetic progressions that are nonempty finite intersections of elements of 
 
 
 
 
 
 B
 
 
 
 
 {\displaystyle {\mathcal {B}}}
 
 will be a base for the topology. Three special cases follow.
The Furstenberg topology,<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> or evenly spaced integer topology,<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> on the set 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
 of integers is obtained by taking as a base the collection of all 
 
 
 
 a
 
 Z
 
 +
 b
 
 
 {\displaystyle a\mathbb {Z} +b}
 
 with 
 
 
 
 a
 ,
 b
 ∈
 
 Z
 
 
 
 {\displaystyle a,b\in \mathbb {Z} }
 
 and 
 
 
 
 a
 >
 0.
 
 
 {\displaystyle a>0.}

The Golomb topology,<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> or relatively prime integer topology,<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> on the set 
 
 
 
 
 
 Z
 
 
 >
 0
 
 
 
 
 {\displaystyle \mathbb {Z} _{>0}}
 
 of positive integers is obtained by taking as a base the collection of all 
 
 
 
 a
 
 N
 
 +
 b
 
 
 {\displaystyle a\mathbb {N} +b}
 
 with 
 
 
 
 a
 ,
 b
 >
 0
 
 
 {\displaystyle a,b>0}
 
 and 
 
 
 
 a
 
 
 {\displaystyle a}
 
 and 
 
 
 
 b
 
 
 {\displaystyle b}
 
 <a href="/facts/Relatively_prime/bnCxCXxs">relatively prime</a>.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> Equivalently,<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> the subcollection of such sets with the extra condition 
 
 
 
 b
 <
 a
 
 
 {\displaystyle b<a}
 
 also forms a base for the topology.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> The corresponding <a href="/facts/Topological_space/v2ut7CbK">topological space</a> is called the Golomb space.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>
The Kirch topology,<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> or prime integer topology,<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> on the set 
 
 
 
 
 
 Z
 
 
 >
 0
 
 
 
 
 {\displaystyle \mathbb {Z} _{>0}}
 
 of positive integers is obtained by taking as a subbase the collection of all 
 
 
 
 p
 
 N
 
 +
 b
 
 
 {\displaystyle p\mathbb {N} +b}
 
 with 
 
 
 
 b
 >
 0
 
 
 {\displaystyle b>0}
 
 and 
 
 
 
 p
 
 
 {\displaystyle p}
 
 prime not dividing 
 
 
 
 b
 .
 
 
 {\displaystyle b.}
 
<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>
Equivalently,<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> one can take as a subbase the collection of all 
 
 
 
 p
 
 N
 
 +
 b
 
 
 {\displaystyle p\mathbb {N} +b}
 
 with 
 
 
 
 p
 
 
 {\displaystyle p}
 
 prime and 
 
 
 
 0
 17</a><a class="footnote-ref" id="fnref:18" href="#fn:18">18</a> A base for the topology consists of all 
 
 
 
 a
 
 N
 
 +
 b
 
 
 {\displaystyle a\mathbb {N} +b}
 
 with relatively prime 
 
 
 
 a
 ,
 b
 >
 0
 
 
 {\displaystyle a,b>0}
 
 and 
 
 
 
 a
 
 
 {\displaystyle a}
 
 <a href="/facts/Squarefree/R3qR9hP1">squarefree</a> (or the same with the additional condition 
 
 
 
 b
 <
 a
 
 
 {\displaystyle b<a}
 
). The corresponding topological space is called the Kirch space.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a>
The three topologies are related in the sense that every open set in the Kirch topology is open in the Golomb topology, and every open set in the Golomb topology is open in the Furstenberg topology (restricted to the subspace 
 
 
 
 
 
 Z
 
 
 >
 0
 
 
 
 
 {\displaystyle \mathbb {Z} _{>0}}
 
). On the set 
 
 
 
 
 
 Z
 
 
 >
 0
 
 
 
 
 {\displaystyle \mathbb {Z} _{>0}}
 
, the Kirch topology is <a href="/facts/Coarser_(topology)/5Q9F2fjf">coarser</a> than the Golomb topology, which is itself coarser that the Furstenberg topology.

<h2 id="properties">Properties</h2>
The Golomb topology and the Kirch topology are <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a>, but not <a href="/facts/Regular_space/gLvRIhRH">regular</a>.<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a><a class="footnote-ref" id="fnref:21" href="#fn:21">21</a>
The Furstenberg topology is Hausdorff and regular.<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a> It is <a href="/facts/Metrizable/BCLdo5Jf">metrizable</a>, but not <a href="/facts/Completely_metrizable/JbJaIzI8">completely metrizable</a>.<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a><a class="footnote-ref" id="fnref:24" href="#fn:24">24</a> Indeed, it is <a href="/facts/Homeomorphic/jMfWg3Mn">homeomorphic</a> to the <a href="/facts/Rational_number/VJQmyY05">rational numbers</a> 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
 with the <a href="/facts/Subspace_topology/NjU6z4Ca">subspace topology</a> inherited from the <a href="/facts/Real_line/6eq42xX5">real line</a>.<a class="footnote-ref" id="fnref:25" href="#fn:25">25</a> Broughan<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a> has shown that the Furstenberg topology is closely related to the <a href="/facts/P-adic_number/SDgDbkLF">p-adic completion</a> of the rational numbers. 
Regarding connectedness properties, the Furstenberg topology is <a href="/facts/Totally_disconnected/lTu5Rvyw">totally disconnected</a>.<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a> The Golomb topology is <a href="/facts/Connected_(topology)/5CUTxfoA">connected</a>,<a class="footnote-ref" id="fnref:28" href="#fn:28">28</a><a class="footnote-ref" id="fnref:29" href="#fn:29">29</a><a class="footnote-ref" id="fnref:30" href="#fn:30">30</a> but not <a href="/facts/Locally_connected/9qhzYyT7">locally connected</a>.<a class="footnote-ref" id="fnref:31" href="#fn:31">31</a><a class="footnote-ref" id="fnref:32" href="#fn:32">32</a><a class="footnote-ref" id="fnref:33" href="#fn:33">33</a> The Kirch topology is both connected and locally connected.<a class="footnote-ref" id="fnref:34" href="#fn:34">34</a><a class="footnote-ref" id="fnref:35" href="#fn:35">35</a><a class="footnote-ref" id="fnref:36" href="#fn:36">36</a>
The integers with the Furstenberg topology form a <a href="/facts/Homogeneous_space/YGebqcJA">homogeneous space</a>, because it is a topological ring — in some sense, the only topology on 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
 for which it is a ring.<a class="footnote-ref" id="fnref:37" href="#fn:37">37</a> By contrast, the Golomb space and the Kirch space are topologically rigid — the only self-<a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphism</a> is the trivial one.<a class="footnote-ref" id="fnref:38" href="#fn:38">38</a><a class="footnote-ref" id="fnref:39" href="#fn:39">39</a>

<h2 id="relation-to-the-infinitude-of-primes">Relation to the infinitude of primes</h2>
Main article: <a href="/facts/Hillel_F%C3%BCrstenberg%27s_proof_of_the_infinitude_of_primes/L1lUKfsX">Hillel Fürstenberg's proof of the infinitude of primes</a>
Both the Furstenberg and Golomb topologies furnish a proof that there are infinitely many <a href="/facts/Prime_number/L20FLkgn">prime numbers</a>.<a class="footnote-ref" id="fnref:40" href="#fn:40">40</a><a class="footnote-ref" id="fnref:41" href="#fn:41">41</a> A sketch of the proof runs as follows:

<ol><li>Fix a prime p and note that the (positive, in the Golomb space case) integers are a union of finitely many <a href="/facts/Residue_class/0wtRa5dU">residue classes</a> modulo p. Each residue class is an arithmetic progression, and thus <a href="/facts/Clopen/7Vrf6PYs">clopen</a>.</li>
<li>Consider the multiples of each prime. These multiples are a residue class (so closed), and the union of these sets is all (Golomb: positive) integers except the <a href="/facts/Unit_(ring_theory)/GJSBu8Y7">units</a> ±1.</li>
<li>If there are finitely many primes, that union is a closed set, and so its complement ({±1}) is open.</li>
<li>But every nonempty open set is infinite, so {±1} is not open.</li></ol>
<h2 id="generalizations">Generalizations</h2>
The Furstenberg topology is a special case of the <a href="/facts/Profinite_topology/0RX9Da9C">profinite topology</a> on a group. In detail, it is the topology induced by the inclusion 
 
 
 
 
 Z
 
 ⊂
 
 
 
 
 Z
 
 ^
 
 
 
 
 
 {\displaystyle \mathbb {Z} \subset {\hat {\mathbb {Z} }}}
 
, where 
 
 
 
 
 
 
 
 Z
 
 ^
 
 
 
 
 
 {\displaystyle {\hat {\mathbb {Z} }}}
 
 is the <a href="/facts/Profinite_integer/hWKKpnwd">profinite integer</a> ring with its profinite topology.
The notion of an arithmetic progression makes sense in arbitrary 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
-<a href="/facts/Module_(mathematics)/jP0LIhFL">modules</a>, but the construction of a topology on them relies on closure under intersection. Instead, the correct generalization builds a topology out of <a href="/facts/Ideal_(ring_theory)/vCvytduX">ideals</a> of a <a href="/facts/Dedekind_domain/tAsvAcaW">Dedekind domain</a>.<a class="footnote-ref" id="fnref:42" href="#fn:42">42</a> This procedure produces a large number of countably infinite, Hausdorff, connected sets, but whether different Dedekind domains can produce homeomorphic topological spaces is a topic of current research.<a class="footnote-ref" id="fnref:43" href="#fn:43">43</a><a class="footnote-ref" id="fnref:44" href="#fn:44">44</a><a class="footnote-ref" id="fnref:45" href="#fn:45">45</a>

<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Hillel_Furstenberg/keLWMrKw">Furstenberg, Harry</a> (1955), "On the infinitude of primes", <a href="/facts/American_Mathematical_Monthly/Te6CHvkg">American Mathematical Monthly</a>, 62 (5), Mathematical Association of America: 353, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2307043">10.2307/2307043</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2307043">2307043</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0068566">0068566</a>.</li>
<li><a href="/facts/Lynn_Arthur_Steen/JFdvAfpc">Steen, Lynn Arthur</a>; <a href="/facts/J._Arthur_Seebach,_Jr./sU4TbeUA">Seebach, J. Arthur Jr.</a> (1995) [1978]. <a href="/facts/Counterexamples_in_Topology/f3eAqwaP">Counterexamples in Topology</a> (<a href="/facts/Dover_Publications/RwjvlSFv">Dover</a> reprint of 1978 ed.). Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-68735-3. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0507446">0507446</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Furstenberg 1955. - Furstenberg, Harry (1955), "On the infinitude of primes", American Mathematical Monthly, 62 (5), Mathematical Association of America: 353, doi:10.2307/2307043, JSTOR 2307043, MR 0068566 <a href="https://doi.org/10.2307%2F2307043" target="_blank">https://doi.org/10.2307%2F2307043</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Golomb, Solomon W. (1959). "A Connected Topology for the Integers". The American Mathematical Monthly. 66 (8): 663–665. doi:10.2307/2309340. ISSN 0002-9890. JSTOR 2309340. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Kirch, A. M. (February 1969). "A Countable, Connected, Locally Connected Hausdorff Space". The American Mathematical Monthly. 76 (2): 169–171. doi:10.1080/00029890.1969.12000163. ISSN 0002-9890. <a href="https://www.tandfonline.com/doi/pdf/10.1080/00029890.1969.12000163" target="_blank">https://www.tandfonline.com/doi/pdf/10.1080/00029890.1969.12000163</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Steen & Seebach, p. 82, counterexample #60, item 1 <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Furstenberg 1955. - Furstenberg, Harry (1955), "On the infinitude of primes", American Mathematical Monthly, 62 (5), Mathematical Association of America: 353, doi:10.2307/2307043, JSTOR 2307043, MR 0068566 <a href="https://doi.org/10.2307%2F2307043" target="_blank">https://doi.org/10.2307%2F2307043</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Steen & Seebach, pp. 80-81, counterexample #58 <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Golomb, Solomon W. (1959). "A Connected Topology for the Integers". The American Mathematical Monthly. 66 (8): 663–665. doi:10.2307/2309340. ISSN 0002-9890. JSTOR 2309340. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Steen & Seebach, pp. 82-84, counterexample #60 <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Golomb, Solomon W. (1959). "A Connected Topology for the Integers". The American Mathematical Monthly. 66 (8): 663–665. doi:10.2307/2309340. ISSN 0002-9890. JSTOR 2309340. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">"The Kirch topology is the same as the prime integer topology". <a href="https://math.stackexchange.com/questions/4460847" target="_blank">https://math.stackexchange.com/questions/4460847</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Steen & Seebach, pp. 82-84, counterexample #60 <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Banakh, Taras; Spirito, Dario; Turek, Sławomir (2021-10-28). "The Golomb space is topologically rigid". Commentationes Mathematicae Universitatis Carolinae. 62 (3): 347–360. arXiv:1912.01994. doi:10.14712/1213-7243.2021.023. ISSN 0010-2628. S2CID 240183836. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Kirch, A. M. (February 1969). "A Countable, Connected, Locally Connected Hausdorff Space". The American Mathematical Monthly. 76 (2): 169–171. doi:10.1080/00029890.1969.12000163. ISSN 0002-9890. <a href="https://www.tandfonline.com/doi/pdf/10.1080/00029890.1969.12000163" target="_blank">https://www.tandfonline.com/doi/pdf/10.1080/00029890.1969.12000163</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Steen & Seebach, pp. 82-84, counterexample #61 <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Banakh, Taras; Stelmakh, Yaryna; Turek, Sławomir (2021-12-01). "The Kirch space is topologically rigid". Topology and Its Applications. 304: 107782. arXiv:2006.12357. doi:10.1016/j.topol.2021.107782. S2CID 219966624. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">"The Kirch topology is the same as the prime integer topology". <a href="https://math.stackexchange.com/questions/4460847" target="_blank">https://math.stackexchange.com/questions/4460847</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Kirch, A. M. (February 1969). "A Countable, Connected, Locally Connected Hausdorff Space". The American Mathematical Monthly. 76 (2): 169–171. doi:10.1080/00029890.1969.12000163. ISSN 0002-9890. <a href="https://www.tandfonline.com/doi/pdf/10.1080/00029890.1969.12000163" target="_blank">https://www.tandfonline.com/doi/pdf/10.1080/00029890.1969.12000163</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Steen & Seebach, pp. 82-84, counterexample #61 <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Banakh, Taras; Stelmakh, Yaryna; Turek, Sławomir (2021-12-01). "The Kirch space is topologically rigid". Topology and Its Applications. 304: 107782. arXiv:2006.12357. doi:10.1016/j.topol.2021.107782. S2CID 219966624. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Steen & Seebach, pp. 82-84, counterexample #60 <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Steen & Seebach, pp. 82-84, counterexample #61 <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">Steen & Seebach, pp. 80-81, counterexample #58 <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">Steen & Seebach, pp. 80-81, counterexample #58 <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">Lovas, R.; Mező, I. (2015). "Some observations on the Furstenberg topological space". Elemente der Mathematik. 70 (3): 103–116. doi:10.4171/EM/283. S2CID 126337479. <a href="http://www.ems-ph.org/doi/10.4171/EM/283" target="_blank">http://www.ems-ph.org/doi/10.4171/EM/283</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Broughan, Kevin A. (August 2003). "Adic Topologies for the Rational Integers". Canadian Journal of Mathematics. 55 (4): 711–723. doi:10.4153/CJM-2003-030-3. ISSN 0008-414X. S2CID 121286344. <a href="https://doi.org/10.4153%2FCJM-2003-030-3" target="_blank">https://doi.org/10.4153%2FCJM-2003-030-3</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">Broughan, Kevin A. (August 2003). "Adic Topologies for the Rational Integers". Canadian Journal of Mathematics. 55 (4): 711–723. doi:10.4153/CJM-2003-030-3. ISSN 0008-414X. S2CID 121286344. <a href="https://doi.org/10.4153%2FCJM-2003-030-3" target="_blank">https://doi.org/10.4153%2FCJM-2003-030-3</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">Steen & Seebach, pp. 80-81, counterexample #58 <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
<li id="fn:28">Steen & Seebach, pp. 82-84, counterexample #60 <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
<li id="fn:29">Golomb, Solomon W. (1959). "A Connected Topology for the Integers". The American Mathematical Monthly. 66 (8): 663–665. doi:10.2307/2309340. ISSN 0002-9890. JSTOR 2309340. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></li>
<li id="fn:30">Szczuka, Paulina (2010-10-01). "The Connectedness of Arithmetic Progressions in Furstenberg's, Golomb's, and Kirch's Topologies". Demonstratio Mathematica. 43 (4): 899–910. doi:10.1515/dema-2010-0416. ISSN 2391-4661. S2CID 122415499. <a href="https://doi.org/10.1515%2Fdema-2010-0416" target="_blank">https://doi.org/10.1515%2Fdema-2010-0416</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></li>
<li id="fn:31">Steen & Seebach, pp. 82-84, counterexample #60 <a href="#fnref:31" class="footnote-back-ref">↩</a></li>
<li id="fn:32">Szczuka, Paulina (2010-10-01). "The Connectedness of Arithmetic Progressions in Furstenberg's, Golomb's, and Kirch's Topologies". Demonstratio Mathematica. 43 (4): 899–910. doi:10.1515/dema-2010-0416. ISSN 2391-4661. S2CID 122415499. <a href="https://doi.org/10.1515%2Fdema-2010-0416" target="_blank">https://doi.org/10.1515%2Fdema-2010-0416</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></li>
<li id="fn:33">Kirch 1969, Theorem 1 - Kirch, A. M. (February 1969). "A Countable, Connected, Locally Connected Hausdorff Space". The American Mathematical Monthly. 76 (2): 169–171. doi:10.1080/00029890.1969.12000163. ISSN 0002-9890. <a href="https://www.tandfonline.com/doi/pdf/10.1080/00029890.1969.12000163" target="_blank">https://www.tandfonline.com/doi/pdf/10.1080/00029890.1969.12000163</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></li>
<li id="fn:34">Steen & Seebach, pp. 82-84, counterexample #61 <a href="#fnref:34" class="footnote-back-ref">↩</a></li>
<li id="fn:35">Kirch, A. M. (February 1969). "A Countable, Connected, Locally Connected Hausdorff Space". The American Mathematical Monthly. 76 (2): 169–171. doi:10.1080/00029890.1969.12000163. ISSN 0002-9890. <a href="https://www.tandfonline.com/doi/pdf/10.1080/00029890.1969.12000163" target="_blank">https://www.tandfonline.com/doi/pdf/10.1080/00029890.1969.12000163</a> <a href="#fnref:35" class="footnote-back-ref">↩</a></li>
<li id="fn:36">Szczuka, Paulina (2010-10-01). "The Connectedness of Arithmetic Progressions in Furstenberg's, Golomb's, and Kirch's Topologies". Demonstratio Mathematica. 43 (4): 899–910. doi:10.1515/dema-2010-0416. ISSN 2391-4661. S2CID 122415499. <a href="https://doi.org/10.1515%2Fdema-2010-0416" target="_blank">https://doi.org/10.1515%2Fdema-2010-0416</a> <a href="#fnref:36" class="footnote-back-ref">↩</a></li>
<li id="fn:37">Broughan 2003, Theorem 2.1 - Broughan, Kevin A. (August 2003). "Adic Topologies for the Rational Integers". Canadian Journal of Mathematics. 55 (4): 711–723. doi:10.4153/CJM-2003-030-3. ISSN 0008-414X. S2CID 121286344. <a href="https://doi.org/10.4153%2FCJM-2003-030-3" target="_blank">https://doi.org/10.4153%2FCJM-2003-030-3</a> <a href="#fnref:37" class="footnote-back-ref">↩</a></li>
<li id="fn:38">Banakh, Taras; Spirito, Dario; Turek, Sławomir (2021-10-28). "The Golomb space is topologically rigid". Commentationes Mathematicae Universitatis Carolinae. 62 (3): 347–360. arXiv:1912.01994. doi:10.14712/1213-7243.2021.023. ISSN 0010-2628. S2CID 240183836. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:38" class="footnote-back-ref">↩</a></li>
<li id="fn:39">Banakh, Taras; Stelmakh, Yaryna; Turek, Sławomir (2021-12-01). "The Kirch space is topologically rigid". Topology and Its Applications. 304: 107782. arXiv:2006.12357. doi:10.1016/j.topol.2021.107782. S2CID 219966624. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:39" class="footnote-back-ref">↩</a></li>
<li id="fn:40">Furstenberg 1955. - Furstenberg, Harry (1955), "On the infinitude of primes", American Mathematical Monthly, 62 (5), Mathematical Association of America: 353, doi:10.2307/2307043, JSTOR 2307043, MR 0068566 <a href="https://doi.org/10.2307%2F2307043" target="_blank">https://doi.org/10.2307%2F2307043</a> <a href="#fnref:40" class="footnote-back-ref">↩</a></li>
<li id="fn:41">Golomb, Solomon W. (1959). "A Connected Topology for the Integers". The American Mathematical Monthly. 66 (8): 663–665. doi:10.2307/2309340. ISSN 0002-9890. JSTOR 2309340. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:41" class="footnote-back-ref">↩</a></li>
<li id="fn:42">Clark, Pete L.; Lebowitz-Lockard, Noah; Pollack, Paul (2018-02-23). "A note on Golomb topologies". Quaestiones Mathematicae. 42 (1): 73–86. doi:10.2989/16073606.2018.1438533. ISSN 1607-3606. S2CID 126371036. <a href="https://www.tandfonline.com/doi/pdf/10.2989/16073606.2018.1438533" target="_blank">https://www.tandfonline.com/doi/pdf/10.2989/16073606.2018.1438533</a> <a href="#fnref:42" class="footnote-back-ref">↩</a></li>
<li id="fn:43">Clark, Pete L.; Lebowitz-Lockard, Noah; Pollack, Paul (2018-02-23). "A note on Golomb topologies". Quaestiones Mathematicae. 42 (1): 73–86. doi:10.2989/16073606.2018.1438533. ISSN 1607-3606. S2CID 126371036. <a href="https://www.tandfonline.com/doi/pdf/10.2989/16073606.2018.1438533" target="_blank">https://www.tandfonline.com/doi/pdf/10.2989/16073606.2018.1438533</a> <a href="#fnref:43" class="footnote-back-ref">↩</a></li>
<li id="fn:44">Spirito, Dario (2019-06-24). "The Golomb topology on a Dedekind domain and the group of units of its quotients". arXiv:1906.09922 [math.GN]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:44" class="footnote-back-ref">↩</a></li>
<li id="fn:45">Spirito, Dario (2019-11-06). "The Golomb topology of polynomial rings". arXiv:1911.02328 [math.GN]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:45" class="footnote-back-ref">↩</a></li>
</ol>

Arithmetic progression topologies open-in-new

Arithmetic progression topologies