*I took a class in point-set topology this fall and we were required to write a final paper for the course. My paper discusses various topologies on the arithmetic progressions and proves theorems regarding their connectedness properties. It has been almost five years since I've posted on Wheels of the Future, and I have been considering bringing it back with the various papers I've been writing, so I thought this was a good opportunity to relaunch the site. Although Wheels of the Future was started to draw analogies to skateboarding from economics and psychology books I had been reading at the time I started the site, it would be cool to take it into a new direction.*

In computer science, I find the graph data structure fascinating. Two graphs can have the same vertices, and even store the same data types in those vertices, but differ greatly in how the vertices are connected. There is a property in graph theory called connectedness, which means between any two vertices in the graph, there is a path or series of edges that connects them. If a graph is not connected, it is disconnected. The connectedness property of a graph is extremely important in applications. For example, suppose that you wanted to find the minimum time it would take to get from one city to another in the United States without paying tolls. You could model each city in the United States as a vertex, create an edge, $(v, w)$, from one city to another if it is possible to get to that city without paying a toll, and then associate with each edge the average time (the cost) it would take to get from the city $v$ to the city $w$. One could use a minimum cost path finding algorithm such as Dijkstra's algorithm to find the minimum time it would take to get from a city, $v$, to a city, $w$, without using tolls. If the graph modeling cities is disconnected, there are cities $a$ and $b$ such that there is no path that connects them without paying tolls and the cost is positive infinity, and it is easy to find what those cities are. There is an even stronger notion of connectedness in graph theory called bi-connectedness, which means the removal of any one vertex and its incident edges from a graph will result in a disconnected subgraph. If a graph is connected, but not bi-connected it must have a cut vertex, which is a vertex that if removed along with its incident edges will result in a disconnected subgraph.

Like graph theory, topology has its own notions of connectedness; a connected space is defined to be a space that has no separation. Similar to how a cut vertex in graph theory causes a connected graph to yield a disconnected subgraph if removed, a cut-point in a connected topological space is a point whose removal causes the resulting space to be disconnected. These analogies between connectedness in topology and connectedness in graph theory are why connectedness is one my favorite topological properties. Another reason I find the topological property of connectedness so fascinating is that there are some extremely interesting and rich proofs showing spaces are connected or not despite how simple the definition of connectedness is.

One of my favorite proofs I did this semester in topology was quite early on in the course. The proof was to show that the collection of arithmetic progressions is a basis for a topology on the integers called Furstenberg's topology. We had just learned about the concept of a basis for a topology, and topology in general was all new to me. I think one of the reasons I enjoyed the proof so much is that any arithmetic progression is a simple set to understand, but the process of showing that if an integer, $x$, lies in two different arithmetic progressions, $A_{1}$ and $A_{2}$, that there must be a third arithmetic progression, $A_{3}$, containing $x$, and then showing $A_{3}$ is contained in $A_{1} \cap A_{2}$ was quite a challenging and stimulating task. After solving this problem it was very cool to find out that Hillel Furstenberg used the arithmetic progressions basis to give a topological proof of the infinitude of the set of prime numbers, $P$. Furstenberg's proof got me extremely excited about prime numbers and it was great to find out that Paulina Szczuka discovered some interesting things about one of my favorite topological properties, connectedness, on $P$ as a subspace of $\Bbb{Z}$ with Furstenberg's topology as well as several other topologies on $\Bbb{Z^{+}}$. This paper describes some of those results and applications.

In Section I, we give basic notation, definitions, and thoerems regarding the topologies we will be using. In Section II are theorems I've reconstructed; the first two theorems were originally proven by Paulina Szczuka [1] in 2009 and examine the connectedness of $P$, and the last theorem is another topological proof of the infinitude of primes, proven by Solomon Golomb [2] in 1959.

**Section I: Notation and basic definitions**

**Notation.** Let $a, b \in \Bbb{Z^{+}}$. $\{an + b\}$ will denote the set $\{an + b \mid n \in \Bbb{Z^{+}} \cup \{0\}\}$ and $\{an\}$ will denote the set $\{an\mid n \in \Bbb{Z^{+}}\}$. $(a, b)$ will denote the greatest common divisor of $a$ and $b$. Let $a \in \Bbb{Z^{+}}$ and $b \in \Bbb{Z}$. $\{az+b\}$ will denote the set $\{az + b\ \mid z \in \Bbb{Z^{+}}\}$.

**Definition 1.1.**
Let $X$ be a space and let $x \in X$. If for each neighborhood, $U$ of $x$ there is some neighborhood $V$ of $x$ contained in $U$ and $V$ is connected, **$X$ is locally connected at $x$**. If for each $x \in X$, $X$ is locally connected at $x$ then $X$ is **locally connected** [3].

**Definition 1.2.**
Let $\mathcal{A}$ $= \{\{az + b \} \mid a \in \Bbb{Z^{+}}, \ b \in \Bbb{Z}\}$. $\mathcal{A}$ is a basis for a topology on $\Bbb{Z}$ called the arithmetic progression topology or **Furstenberg's topology**. When $\Bbb{Z}$ is given Furstenberg's topology, we will denote it by $\Bbb{Z}_{F}$.

**Remark 1.3.** Hillel Furstenberg used $\mathcal{A}$ to show that there are infinitely many primes in 1955 while an undergraduate at Yashiva University where he majored in mathematics and divinity. Furstenberg currently resides in Israel and retired from a professorship at Hebrew University in Jerusalem in 2003 [4].

**Theorem 1.4. (Dirichlet's theorem on arithmetic progressions).** Let $\{an + b\} \in$ $\mathcal{A}$. If $a$ and $b$ are positive integers and coprime, there are infinitely many prime numbers in $\{an + b\}$.

**Definition 1.5.** Let $\mathcal{B}$ $= \{\{an + b \} \mid a, b \in \Bbb{Z}^{+},\ (a, b) = 1\}$. $\mathcal{B}$ is a basis for a topology on $\Bbb{Z}^{+}$ called **Golomb's topology**. When $\Bbb{Z}^{+}$ is given Golomb's topology, we will denote it by $\Bbb{Z}^{+}_{G}$.

**Remark 1.6.**
In addition to giving a proof for the infinitude of primes with the space $\Bbb{Z}^{+}_{G}$ in 1959, Golomb also proved that the space $\Bbb{Z}^{+}_{G}$ is Hausdorff and connected in the same paper. Although, Golomb received his PhD in mathematics from Harvard where he focused on number theory and topology, he became interested in communications theory while working at the Glen L. Martin company and has made great contributions to information theory. For fun Golomb creates mathematical games and is the creator of pentomino, the inspiration for the classic video game, Tetris. He is currently a professor of electrical engineering at USC.

**Definition 1.7.** Let $x$ be an integer. $x$ is said to be **square-free** if the only perfect square $x$ is divisible by is $1$.

**Definition 1.8.** Let $\mathcal{B}' = \{\{an + b \} \mid a, b \in \Bbb{Z}^{+},\ (a, b) = 1,\ b < a,\ a\ \text{ is square-free}\}$. $\mathcal{B}$ is a basis for a topology on $\Bbb{Z}^{+}$ called **Kirch's topology**. When $\Bbb{Z}^{+}$ is given Kirch's topology, we will denote it by $\Bbb{Z}^{+}_{K}$. It should be noted that Golomb's topology is strictly finer than Girch's topology.

**Section II: Theorems**

**Theorem 2.1 (Szczuka).** The set of all prime numbers is disconnected in Golomb's and Kirch's topologies.

*proof*: Let $P$ be a subspace of $\Bbb{Z}^{+}_{K}$. $$\text{ Suppose }A_{1} = \{3n + 2\}\cup\{5n + 2\}\cup\{5n + 3\}\cup\{5n + 1\} \text{ and } B_{1} = \{15n + 4\}.$$ $A_{1}$ is a union of open sets in $\mathcal{B}'$, and $B_{1}$ is an element of $\mathcal{B}'$. Thus $A_{1}$ and $B_{1}$ are open in $\Bbb{Z}^{+}_{K}$. $A_{1}$ and $B_{1}$ are disjoint and both have infinitely many primes by Dirichlet's theorem, as $\{3n + 2\}\subset A_{1}$ and $\{15n + 4\}\subset B_{1}$ where $2, 3$ are coprime and $15, 4$ are coprime. Thus $A = A_{1}\cap P$ is non-empty and $B = B_{1}\cap P$ is non-empty since there are infinitely many primes in both $A_{1}$ and $B_{1}$. $A$ and $B$ are disjoint since $A_{1}$ and $B_{1}$ are disjoint. Note that $A$ and $B$ are open sets in $P$.

We will show that $A\cup B = P$ and $P$ is disconnected. $$\Bbb{Z}^{+} = A_{1} \cup B_{1} \cup \{15n + 9\} \cup \{15n + 10\}\cup\{15n\} \text{ which means }$$ $$P \cap \Bbb{Z}{+} = P\cap (A_{1}\cup B_{1}\cup\{15n + 9\}\cup\{15n + 10\}\cup\{15n\}) = P$$ Since $P\cap\{15n + 9\}$, $P\cap\{15n + 10\}$, and $P\cap\{15n\}$ are all empty sets, $P = (P\cap A_{1})\cup(P\cap B_{1})$ and $P = A\cup B$. Since $A$ and $B$ are non-empty and disjoint, they form a separation of $P$, and $P$ is disconnected as a subspace of $\Bbb{Z}^{+}_{K}$. Since the topology on $\Bbb{Z}^{+}_{G}$ is strictly finer than the topology on $\Bbb{Z}^{+}_{K}$, $P$ is disconnected as a subspace of $\Bbb{Z}^{+}_{G}$ [1].

**Theorem 2.2 (Szczuka).** The set of all prime numbers is locally connected in Furstenberg's topology.

*proof*: Let $P$ be a subspace of $\Bbb{Z}_{F}$. Let $p \in P$, and let $U$ be a neighborhood of $p$. There must be some set $\{az + b\} \in \mathcal{A}$ which is open in $P$, contains $p$, and is contained in $U$. Let $b = 0$ such that $\{pz\} \in \mathcal{A}$ and the only prime number in $\{pz\}$ is $p$. Thus $P\cap \{pz\} = \{p\}$, which is connected as a subspace of $P$ and also is an open set in $P$. Since $p \in \{p\} \subset U$, $P$ is locally connected at $p$, and hence $P$ is locally connected, as $P$ is locally connected for each $p \in P$ [1].

**Theorem 2.3 (Szczuka).**: The set of all prime numbers is not locally connected in Kirch's or Golomb's topologies.

*proof*: Let $P$ be a subspace of $\Bbb{Z}^{+}_{K}$. Suppose that $P$ is locally connected. $\{3n + 2\} \in \mathcal{B'}$ and $\{3n + 2\}\cap P$ is a non-empty open set in $P$ since it contains $2$. Since $P$ is locally connected there is a connected neighborhood of $2$, $H$ contained in $\{3n + 2\}\cap P$. Since $H$ is open, there is some set $\{an + b\}\cap P$ in $P$'s basis containing $2$ and contained in $H$. Since $\{an +b\} \in \mathcal{B'}$, $\{an +b\}$ must contain infinitely many prime numbers and $\{an + b\}\cap P$ contains infinitely many prime numbers.

Choose $p_{1} \in \{3n + 2\}\cap P$ such that $p_{1}$ does not equal $2$. $\{3n+1\} \in \mathcal{B'}$ and has infinitely many prime numbers, and we pick $p \in \{3n + 1\}$ such that $p$ is greater than $p_{1}$. Note that $p \not\in \{3n + 2\}$. $$\bigcup\limits_{k=1}^{p}\{pn + k\} = \Bbb{Z}^{+} \text{ and } P \cap \Bbb{Z}^{+} = P \text{ and } P\cap\bigcup\limits_{k=1}^{p}\{pn + k\} = P.$$ $$\{pn + p\}\cap\{3n + 2\} \text{ is empty, so } P\cap\{3n +2\} = P\cap(\bigcup\limits_{k=1}^{p-1}\{pn + k\})\cap\{3n + 2\}.$$ $$\text{Since } p_{1} < p\text{, }p_{1} \in \{pn + p_{1}\}\subset\bigcup\limits_{k=1}^{p - 1}\{pn + k\}\text{ and } 2 \in \{pn + 2\}\cap\bigcup\limits_{k=1}^{p - 1}\{pn + k\}.$$ Let $A = \{pn + p_{1}\}\cap\{3n + 2\}$. If $p_{1} = p - 1$ then let $$B = P\cap(\bigcup\limits_{k=1}^{p-2}\{pn + k\})\cap\{3n + 2\}$$ Otherwise, let $$B = P\cap ((\bigcup\limits_{k=1}^{p_{1} - 1}\{pn + k\} )\cup (\bigcup\limits_{j=p_{1} + 1}^{p - 1}\{pn + j\} ))\cap\{3n + 2\}$$ $A$ does not intersect $B$ and $p_1 \in A$ and $2 \in B$, so $A$ and $B$ are non-empty. Moreover, for each $k \in \Bbb{Z}^{+}$ such that $1 \leq k < p$, $\{pn + k\}\cap P$ is open in $P$ which means both $A$ and $B$ are open in $P$.

$A\cup B = P\cap\bigcup\limits_{k=1}^{p-1}(\{pn + k\})\cap\{3n + 2\} = P\cap\{3n + 2\}$ and $H \subset A \cup B$. Thus $A \cap H$ and $B \cap H$ are open in $H$ since $A$ and $B$ are open in $P$, and $A \cap H$ and $B \cap H$ are disjoint in $H$ since $A$ and $B$ are disjoint in $P$. $A \cap H$ contains $p_{1}$ since both $A$ and $H$ contain $p_{1}$ and $B \cap H$ contains $2$ since both $H$ and $B$ contain $2$. Thus $A \cap H$ and $B \cap H$ are non-empty and form a separation of $H$, a contradiction. Thus $P$ is not locally connected as a subspace of $\Bbb{Z}^{+}_{K}$. Since the topology on $\Bbb{Z}{+}_{G}$ is strictly finer than the topology on $\Bbb{Z}^{+}_{K}$, $P$ is not locally connected as a subspace of $\Bbb{Z}^{+}_{G}$ [1].

**Theorem 2.4 (Euclid).**: There are infinitely many prime numbers.

*proof*: Consider $\Bbb{Z}^{+}$ with Golomb's topology. Let $p \in P$. $\{pn\}$ is closed, as $\{pn\}$'s complement is $\bigcup\limits_{k=1}^{p-1}\{pn + k\}$ and open in $\Bbb{Z}^{+}_{G}$ since for each $\{pn + k\}$ such that $k \in \Bbb{Z}^{+} \text{ and } 1 \leq k \leq p-1$ is in $\mathcal{B}'$. Suppose $P$ is finite. $\Bbb{Z}^{+} - \{1\} = \bigcup\limits_{p \in P}\{pn\}$ must be closed, as it the the union of a finite number of closed sets in $\Bbb{Z}^{+}_{G}$. $\Bbb{Z}^{+} - \{1\}$'s complement is $\{1\}$, which is not open in $\Bbb{Z}^{+}_{G}$, as it is not infinite or empty. Thus $\Bbb{Z}^{+} - \{1\}$ cannot be closed and hence $\bigcup\limits_{p \in P}\{pn\}$ must be infinite and there must be infinitely many prime numbers [2].

**References:**

[1] Paulina Szczuka, “The connectedness of arithmetic progressions in furstenbergs, golombs,and kirchs topologies,” Demonstratio Math, vol. 43, pp. 899–909, 2010.

[2] Solomon Golomb, “A connected topology for the integers,” The American Mathematical Monthly, vol. 66, pp. 663– 665, 1959.

[3] James R. Munkres, Topology. Prentice Hall, Inc., 2000.

[4] JJ O’Connor and EF Robertson, Hillel Furstenberg, 2010. http://www-history.mcs.st-andrews.ac.uk/
Biographies/Furstenberg.html.

[5] Wikipedia, Solomon Golomb, 2015. https://en.wikipedia.org/wiki/Solomon_W._Golomb.