Thread: Can open balls and pairwise intersections of balls generate a topology?

Hi,

I have a question. Can the following topology base generate a topology(a family of sets)?:

X - finite set of points, d - metric
B - open ball in X
r - ray, const value

Additionally, how could be the cover of a family of sets in topology realized ?

Thans in advance,
algorytmus

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

Well if you mean that , then it still generates the discrete topology.