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I have to show clearly. If $a$ and $b$ are element of $\mathbb{Z}$ (whole numbers) and $b>0$
Is the family $$B=\{N_{a,b} : a,b\in \mathbb{Z} , b > 0\}$$ for $N_{a,b}=\{{a+kb : k\in \mathbb{Z}}\}$ a base for some topology on $\mathbb{Z}$ ?

I think that the answer is yes and I know two properties that are needed to be a base for topology. However, I cannot apply for this question.

Any answer will be helpfull for me, thanks a lot.

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    Which of the two properties in the definition (of a base for a topology) are you having trouble with? – 2017-01-08
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    1.The base elements cover X. 2.Let B1, B2 be base elements and let I be their intersection. Then for each x in I, there is a base element B3 containing x and contained in I – 2017-01-08
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    Good! Which one of them do you need help with? Or do you need help with both? – 2017-01-08
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    I need help with both. I can see that both is true but I cannot show. @zipirovich – 2017-01-08

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Note that each $N_{a,b}$ is an arithmetic sequence (progression) with a common difference $b$ extending infinitely in both directions. How, hints for both properties.

  1. The first property is essentially trivial. You need to show that any $a\in\mathbb{Z}$ belongs to at least one of those sets. Can you see what you should write in place of question mark in "$a\in N_{?,b}$" to see that it's true?

  2. This is based on an observation that the intersection of two arithmetic progressions, assuming it's non-empty, is also an arithmetic progression. More specifically, if $B_1=N_{a_1,b_1}$ and $B_2=N_{a_2,b_2}$, and if $x\in B_1\cap B_2$ (thus guaranteeing that it's non-empty), then $x\in N_{x,d}$ for some appropriate $d$ (which depends on $b_1,b_2$).

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    1. I think now I may write ?=a because N_a,b contains the element a. Also N_a-kb,b also contains the element a – 2017-01-08
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    2. Also if I choose d = lcm(b1,b2) will it work? It is clear that the set will be in the intersection. Also x is element of this set? @zipirovich – 2017-01-08
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    Yes, that's right! – 2017-01-08
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    Related to this basis: Hillel Furstenberg's proof of the infinitude of primes https://en.wikipedia.org/wiki/Furstenberg's_proof_of_the_infinitude_of_primes – 2017-01-08
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    Thank you very much! Now, after learning the answer, I start to think that was an easy question. Thank you for spending your valuable time for my question. Best wishes. @zipirovich – 2017-01-08
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    Thank you very much for this usefull link and spending your valuable time for me. Best wishes. @Alone – 2017-01-08