Exercise 2.1.3 (Convergence in cofinite topology)

Question

Let $\mathcal{T}$ be the cofinite topology on the set $\mathbb{Z}$ of integers. Show that the sequence $\{1,2,3,\ldots\}$ converges in $(\mathbb{Z},\mathcal{T})$ to each point of $\mathbb{Z}$. Describe the convergent sequences in $(\mathbb{Z},\mathcal{T})$.

Elu Thingol · Accepted Answer

Recall that to demonstrate that the sequence $(x_n)_{n \in \mathbb{N}}$ converges to an integer $m \in \mathbb{Z}$ is to show that for every open neighbourhood $U$ of $x$ there exists an index $N \in \mathbb{N}$ such that the sequence $(x_n)_{n \in \mathbb{N}}$ eventually lies in $U$, i.e., $x_n \in U$ for all $n > N$.

\begin{proof}
Let $m \in \mathbb{Z}$ be an arbitrary integer and let $U$ be some arbitrary open neighbourhood of $m$. By definition, $\mathbb{Z} \setminus U$ is finite; on other words, only finitely many integers are not contained in $U$. Thus, taking $N:=\max \mathbb{Z} \setminus U$, we see that the sequence $\{1,2,3,\ldots\}$ is contained in $U$ starting with $i > N$. Since our choice of $U$ was arbitrary, $\{1,2,3,\ldots\}$ converges to $m$. Since our choice of $m$ was arbitrary, we see that $\{1,2,3,\ldots\}$ converges to every point of $\mathbb{Z}$.
\end{proof}

More generally, a sequence $(x_n)_{n \in \mathbb{N}}$ of integers in the cofinite space converges to an integer $m$ if and only if each integer $j 
eq m$ occurs in the sequence only a finite number of times (otherwise we could always take an infinite set around $m$ without the element of the sequence $j 
eq m$ which occurs infinitely). Thus, the convergent sequences in $(\mathbb{Z},\mathcal{T})$ are exactly those for which at most one value $m$ occurs infinitely often.

Exercise 2.1.3 (Convergence in cofinite topology)

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Exercise 2.1.3 (Convergence in cofinite topology)

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