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I'm being asked to prove that $\mathcal{A}=\left\lbrace A_{a,b}:a,b\in\mathbb{Z}\right\rbrace$ is a basis for a topology on $\mathbb{Z}$, being: $$A_{a,b}=\left\lbrace a+nb:n\in\mathbb{Z}\right\rbrace=\left\lbrace ...,a-2b,a-b,a,a+b,a+2b,...\right\rbrace$$

Obviously every $a\in \mathbb{Z}$ belongs to a basis element, for example $A_{a,b}$, for any $b$. I'm having difficulties though, proving that if two of those sets $A_{a,b}$ and $A_{a',b'}$ share an element $x$, then there exists another $A_{c,d}$, such that $x\in A_{c,d}$ and $A_{c,d}\subset A_{a,b},A_{a',b'}$. Any hint?

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You might be interested in the co-prime integer topology on $\mathbb Z$—an example of a Hausdorff topology which is not Urysohn. – kahen Mar 22 '13 at 20:33
@kahen Thanks, I'll read about it. – MyUserIsThis Mar 22 '13 at 20:57
up vote 3 down vote accepted

Hint: Try $d = \operatorname{lcm}(b,b')$ or $d = bb'$ - either should work fine. The idea being that we're looking for things that are in a list of step size of $b$ (so to say) and in another list of step size $b'$.

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We need not to consider $c$? It can be anything? – Idonknow Oct 28 '13 at 4:12

As a hint, sets with large values of the second argument are smaller, as they include fewer numbers.

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Hint: Chinese Remainder Theorem.

If $\rm x\equiv a\bmod b,\bar{a}\bmod\bar{b}$, then pick $\rm d:=lcm(b,\bar{b})$ and $\rm c$ a residue of $\rm x$ mod $\rm d$.

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If $z\in D_1= A_{a_1,b_1}\in A$ and $z\in D_2=a_{a_2,b_2}\in A,$ then $D_1=A_{z,b_1}$ and $D_2=A_{z,b_2}$ so $z\in D_3\subset (D_1\cap D_2)$ where $D_3=A_{z,(b_1 b_2)}\in A.$

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