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

1. 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

2. Re: Can open balls and pairwise intersections of balls generate a topology?

A family $\mathcal{F}$ of sets generates a topology on $X$ just in case:

(1) $\mathcal{F}\subseteq 2^X$;
(2) $x\in X$ $\implies$ $\exists B\in\mathcal{F}$ such that $x\in B$;
(3) $x\in B_1\cap B_2$ for $B_1,B_2\in\mathcal{F}$ $\implies$ $\exists B_3\in\mathcal{F}$ such that $x\in B_3\subseteq B_1\cap B_2$.

Your collection satisfies these requirements. In fact, notice that the second term of the union is superfluous, and that $\mathcal{B}=\{B_d(x,r):x\in X,r>0\}$. Given that $X$ is finite, this means $\mathcal{B}=2^X\setminus\emptyset$, i.e. it generates the discrete topology.

I'm not sure what you mean by realizing a cover of a family of sets though.

3. Re: Can open balls and pairwise intersections of balls generate a topology?

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

$\mathcal{B}$={{B_a},{B_b},{B_c},{B_a,B_b},{B_b,B_c}}

,
for

it is:

$\mathcal{B}$={{B_a},{B_b},{B_c}}

B_x - a ball for x

4. Re: Can open balls and pairwise intersections of balls generate a topology?

Well if you mean that $\mathcal{B}=\{\{x\}:x\in X\}$, then it still generates the discrete topology.