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Let $\mathcal T$ be a topology on $\mathbb Z:$ $\mathcal T\subset\mathcal P(\mathbb Z)$ with $\emptyset$ and all the unions of $S(a,b)=\{an+b|n\in \mathbb Z\}$ for $a\neq 0$ prove that $\mathcal T$ is topology on the integers.

I know that I should show:

$\ \ \ $1) $\emptyset$ are in $\mathcal T$.

$\ \ \ $2) Any union of elements of $T$ belongs to $\mathcal T$.

$\ \ \ $3) Any finite intersection of elements of $\mathcal T$ belongs to $\mathcal T$

Have no idea how to start

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  • Well, would you agree that the first two requirements are obvious? It's the third one that might need a bit of work. Showing that $S(a_1,b_1)\cap S(a_2,b_2)$ belongs to $\mathcal{T}$ would be a good next step. – Harald Hanche-Olsen Mar 28 '17 at 18:49
  • Well $\emptyset$ is given by definition, unions syould be obvious by definition, so you just have to check the last part – Maxime Ramzi Mar 28 '17 at 18:50
  • You wish to show that the set of all $S(a,b) $ with $a\ne 0$ is a base for a topology on $\mathbb Z.$... A family $F$ of subsets of set $X$ is a base for a topology on $X$ iff (1)$;\cup F=X$ (Every $p\in X$ belongs to at least one member of $F$) and (2) if $f_1,f_2\in F$ and $p\in f_1\cap f_2$ then there exists $f_3 \in F$ with $p\in f_3\subset f_1\cap f_2$...... If $p\equiv b_i \pmod {a_i}$ for $i=1,2 $, with positive $a_1,a_2,$ what can you say about $p$ modulo lcm $(a_1,a_2)$ and $S(lcm (a_1,a_2),p)$? – DanielWainfleet Mar 28 '17 at 22:54

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