(a) The followings are the basis elements of B.

B1={1,2,3,.................}

B2={2,3,4,...............}

B3={3,4,5,............}

........

Bn={n, n+1, n+2,,,,}

...........

To check B is a basis for Z+, you first need to check the union of the above elements are indeed Z+.

Second, you need to verify that for each x in the interesection of the basis elements, there exists a basis element containing x and contained in the intersection.

For instance, consider the intersection of B1 and B2. For each x in the section, you can find a basis element that containing x.

(b) Pick two numbers, for instance, 3 and 5. Check open sets containing 3 & 5 respectively. Those open sets cannot be chosen disjointly.

(c) Every open set containing a number in Z+ contains all but finite numbers of the above sequence in your topology. Thus, it converges everywhere in Z+.