$X=\mathbb{Z}$,$\mathfrak B:= \{a+b\mathbb{Z}:a \in \mathbb{Z}, b \in \mathbb{N}\}$ is a basis of a topology on $X$?

Question

$X=\mathbb{Z}$

$$\mathfrak B:= \{a+b\mathbb{Z}:a \in \mathbb{Z}, b \in \mathbb{N}\}$$

I need to show that $\mathfrak{B}$ is as basis of a topology on $X$, but I have no clue which topology has $\mathfrak{B}$ as a basis.

The hint I received is $a+b \mathbb{Z}=x+b \mathbb{Z} \ \forall x \in a + b \mathbb{Z}$ but that doesn't really help me either.

Answer 1

Another hint:

If you must check whether any collection $\mathfrak B$ serves as basis of a topology then you can do that without knowing which topology.

You only have to check:

• Do the elements of $\mathfrak B$ cover $X$?
• If $B_1,B_2\in\mathfrak B$ and $x\in B_1\cap B_2$ then does there always exist some $B\in\mathfrak B$ with $x\in B\subseteq B_1\cap B_2$?

If and only if both questions get the answer "yes" then $\mathfrak B$ can be classified as basis of topology.

The topology will be the collection of unions of elements of $\mathfrak B$.