Question

Recall that the arithmetic progression topology on $\mathbb{Z}$ is generated by the basis $\mathcal{B}=\left\{A_{a, b} \mid a, b \in \mathbb{Z}, b \neq 0\right\}$, where each $$ A_{a, b}=\{\ldots, a-2 b, a-b, a, a+b, a+2 b, \ldots\} $$ is an arithmetic progression. Determine whether or not $\mathbb{Z}$ is compact in this topology.

   Recall that the arithmetic progression topology on $\mathbb{Z}$ is generated by the basis $\mathcal{B}=\left\{A_{a, b} \mid a, b \in \mathbb{Z}, b \neq 0\right\}$, where each
$$
A_{a, b}=\{\ldots, a-2 b, a-b, a, a+b, a+2 b, \ldots\}
$$
is an arithmetic progression. Determine whether or not $\mathbb{Z}$ is compact in this topology.

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Introduction to Topology: Pure and Applied

Introduction to Topology: Pure and Applied

Colin Adams, Robert… 1st Edition

Chapter 7, Problem 7

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Step 1

A topological space is compact if every open cover has a finite subcover. This means that if we have a collection of open sets that cover the space, we can find a finite number of those sets that still cover the space. Show more…

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Recall that the arithmetic progression topology on $\mathbb{Z}$ is generated by the basis $\mathcal{B}=\left\{A_{a, b} \mid a, b \in \mathbb{Z}, b \neq 0\right\}$, where each $$ A_{a, b}=\{\ldots, a-2 b, a-b, a, a+b, a+2 b, \ldots\} $$ is an arithmetic progression. Determine whether or not $\mathbb{Z}$ is compact in this topology.

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