A family of sets generates a topology on just in case:
(1) ;
(2) such that ;
(3) for such that .
Your collection satisfies these requirements. In fact, notice that the second term of the union is superfluous, and that . Given that is finite, this means , i.e. it generates the discrete topology.
I'm not sure what you mean by realizing a cover of a family of sets though.
Thank you for help!
Could you correct me if I am wrong in the following:
Let me illustrate examples geometrically.
Given X={a, b, c}
the topological base is
={{B_a},{B_b},{B_c},{B_a,B_b},{B_b,B_c}}
,
for
it is:
={{B_a},{B_b},{B_c}}
B_x - a ball for x